Kinematics of Rigid Body

Kinematics of Rigid Body

Contents

Describe the following Motion Curves i. Displacement-time curve ii. Velocity-time curve iii. Acceleration-time curve 2

Recall the following Motion i. Motion with Uniform Velocity ii. Motion with Uniform Acceleration iii. Motion with variable Acceleration 4

Describe the concept of relative velocity in linear motion 5

Define the following terms: i. Relative distance ii. Resultant velocity 6

Derive the expression for acceleration during circular motion 7

Describe the circular motion on level ground 8

Derive the relation between the angle of banking and design speed 10

Recall the concept of Skidding and Overturning on banked road 12

Describe the Angular Motion 13

Derive a relation between angular motion and linear motion 14

Derive the expressions for: i. Uniformly angular velocity ii. Uniformly accelerated rotation 15

Explain the kinetics of rigid body rotation 16

Define the following terms related to projectile i. Velocity of projection ii. Angle of projection iii. Trajectory iv. Horizontal range v. Time of flight 17

Describe the motion of body projected horizontally 18

Describe the inclined projection on level ground 19

Describe the projection at different levels with the point of projection and point of strike 21

Describe the following Motion Curves i. Displacement-time curve ii. Velocity-time curve iii. Acceleration-time curve

  1. Displacement-Time Curve: The displacement-time curve is a graph that shows the relationship between the displacement of an object and the time elapsed during its motion. Displacement is the change in position of an object and can be represented by the straight-line distance between the initial and final position of the object. The displacement-time curve shows how the displacement of an object changes with time.

The slope of the displacement-time curve represents the velocity of the object, while the area under the curve represents the distance travelled by the object. The shape of the displacement-time curve can indicate the type of motion of the object, such as linear, circular, or oscillatory motion.

  1. Velocity-Time Curve: The velocity-time curve is a graph that shows the relationship between the velocity of an object and the time elapsed during its motion. Velocity is the rate of change of displacement and can be represented by the slope of the displacement-time curve. The velocity-time curve shows how the velocity of an object changes with time.

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The slope of the velocity-time curve represents the acceleration of the object, while the area under the curve represents the change in velocity of the object. The shape of the velocity-time curve can indicate the type of motion of the object, such as linear, circular, or oscillatory motion.

  1. Acceleration-Time Curve: The acceleration-time curve is a graph that shows the relationship between the acceleration of an object and the time elapsed during its motion. Acceleration is the rate of change of velocity and can be represented by the slope of the velocity-time curve. The acceleration-time curve shows how the acceleration of an object changes with time.

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The shape of the acceleration-time curve can indicate the type of motion of the object, such as linear, circular, or oscillatory motion. The acceleration-time curve can also be used to determine the forces acting on an object and the direction of the forces, as the acceleration of an object is proportional to the net force acting on it.

Recall the following Motion i. Motion with Uniform Velocity ii. Motion with Uniform Acceleration iii. Motion with variable Acceleration

  1. Motion with Uniform Velocity: Motion with uniform velocity is a type of motion in which an object moves at a constant speed in a straight line. The velocity of an object in uniform velocity motion is constant and does not change with time. This means that the displacement-time curve for an object in uniform velocity motion is a straight line with a constant slope, representing the constant velocity of the object.

The velocity-time curve for an object in uniform velocity motion is a horizontal line, indicating that the velocity is constant and does not change with time. The acceleration-time curve for an object in uniform velocity motion is a horizontal line at zero, indicating that the acceleration is zero and the velocity is constant.

  1. Motion with Uniform Acceleration: Motion with uniform acceleration is a type of motion in which an object moves in a straight line and experiences a constant acceleration. The acceleration of an object in uniform acceleration motion is constant and does not change with time. This means that the velocity-time curve for an object in uniform acceleration motion is a straight line with a constant slope, representing the constant acceleration of the object.

The displacement-time curve for an object in uniform acceleration motion is a parabolic curve, indicating that the displacement increases at a constant rate with time. The acceleration-time curve for an object in uniform acceleration motion is a horizontal line, indicating that the acceleration is constant and does not change with time.

  1. Motion with Variable Acceleration: Motion with variable acceleration is a type of motion in which an object moves in a non-uniform manner, experiencing changes in acceleration. The acceleration of an object in variable acceleration motion changes with time, resulting in a curved velocity-time curve and a curved acceleration-time curve.

The velocity-time curve for an object in variable acceleration motion shows how the velocity changes with time, with the slope of the curve representing the acceleration at any given time. The acceleration-time curve for an object in variable acceleration motion shows how the acceleration changes with time. The shape of the acceleration-time curve can indicate the type and direction of the forces acting on the object, as well as the nature of the motion.

Describe the concept of relative velocity in linear motion

The concept of relative velocity in linear motion refers to the velocity of one object relative to another object in a linear or straight-line motion. In other words, it is the velocity of one object as seen from another object. The relative velocity between two objects can be determined by subtracting the velocity of one object from the velocity of the other object.

For example, consider two objects A and B moving in the same direction, with object A moving at a velocity of 5 m/s and object B moving at a velocity of 10 m/s. The relative velocity of object B with respect to object A is 10 m/s – 5 m/s = 5 m/s. This means that if you were standing on object A, you would see object B moving away from you at a velocity of 5 m/s.

Similarly, if two objects A and B are moving in opposite directions, the relative velocity can be determined by adding the velocities of the two objects. For example, if object A is moving at a velocity of 5 m/s to the right and object B is moving at a velocity of 10 m/s to the left, the relative velocity of object B with respect to object A is 10 m/s + 5 m/s = 15 m/s to the left. This means that if you were standing on object A, you would see object B moving away from you at a velocity of 15 m/s to the left.

The concept of relative velocity is useful in many applications, including in the analysis of collision and impact problems, in the calculation of relative speeds in fluid dynamics, and in the determination of relative velocities in projectile motion.

Define the following terms: i. Relative distance ii. Resultant velocity

i. Relative distance: The relative distance between two objects is the distance between the two objects as seen from one of the objects. In other words, it is the distance between two objects from the reference frame of one of the objects. The relative distance is a useful concept in determining the relative position of one object with respect to another object, especially in situations where both objects are in motion.

ii. Resultant velocity: The resultant velocity is the net or total velocity of an object when it is in motion. In other words, it is the vector sum of all the velocities that the object has in different directions. The resultant velocity is a useful concept in determining the overall speed and direction of an object that is in motion. For example, consider an object that is moving in two perpendicular directions with velocities v1 and v2. The resultant velocity of the object is the vector sum of v1 and v2, and it gives the overall speed and direction of the object.

In many applications, the concept of resultant velocity is useful in determining the relative velocity between two objects in motion, especially when the two objects are moving in different directions. The relative velocity between two objects can be determined by subtracting the velocity of one object from the velocity of the other object, or by finding the resultant velocity of the two objects with respect to a common reference frame.

Derive the expression for acceleration during circular motion

The expression for acceleration during circular motion can be derived using the following steps:

  1. Definition of Circular Motion: Circular motion is a type of motion in which an object moves in a circular path around a fixed center. The speed of the object is constant, but its direction is constantly changing.
  2. Centripetal Force: The force that keeps an object moving in a circular path is called centripetal force. It is directed towards the center of the circle and is equal to the mass of the object multiplied by its acceleration.
  3. Centripetal Acceleration: The acceleration experienced by an object moving in a circular path is called centripetal acceleration, which is directed towards the center of the circle. It is given by the equation:

a = v2 / r

where “a” is the centripetal acceleration, “v” is the velocity of the object, and “r” is the radius of the circular path.

  1. Relation between Centripetal Force and Centripetal Acceleration: Using Newton’s second law, we can relate the centripetal force to the centripetal acceleration. The equation is given by:

F = m * a

where “F” is the centripetal force, “m” is the mass of the object, and “a” is the centripetal acceleration.

  1. Final Expression for Acceleration during Circular Motion: Combining the above equations, we can obtain the expression for the acceleration during circular motion as:

a = v2 / r = m * F / m = F / m

where “a” is the acceleration during circular motion, “v” is the velocity of the object, “r” is the radius of the circular path, “m” is the mass of the object, and “F” is the centripetal force.

This expression provides us with a way to calculate the acceleration experienced by an object moving in a circular path given its velocity and the radius of the circular path.

Describe the circular motion on level ground

Circular motion on level ground refers to the motion of an object moving in a circular path on a flat, horizontal surface. This type of motion is commonly seen in everyday life, for example, in the motion of a wheel spinning on a road or a ball rolling along the ground in a circular path.

  1. Centripetal Force: The force that keeps an object moving in a circular path is known as centripetal force. This force acts towards the center of the circle and is responsible for changing the direction of the object’s velocity while keeping its speed constant. The centripetal force is given by the equation:

F = m * a

where “F” is the centripetal force, “m” is the mass of the object, and “a” is the centripetal acceleration.

  1. Centripetal Acceleration: The acceleration experienced by an object moving in a circular path is known as centripetal acceleration, and it is directed towards the center of the circle. It can be calculated using the equation:

a = v2 / r

where “a” is the centripetal acceleration, “v” is the velocity of the object, and “r” is the radius of the circular path.

  1. Frictional Force: In the case of circular motion on level ground, the frictional force also plays an important role. Frictional force acts in the direction opposite to the motion of the object and provides the necessary centripetal force to keep the object moving in a circular path.
  2. Net Force: The net force acting on an object in circular motion on level ground is given by the difference between the centripetal force and the frictional force. If the net force is greater than the frictional force, the object will move in a larger circular path, while if the net force is less than the frictional force, the object will move in a smaller circular path.
  3. Example: Consider the example of a ball rolling on a level ground in a circular path. The friction between the ball and the ground provides the necessary centripetal force to keep the ball moving in a circular path. The velocity of the ball and the radius of the circular path can be calculated using the equations for centripetal force and centripetal acceleration.

In conclusion, circular motion on level ground refers to the motion of an object moving in a circular path on a flat, horizontal surface, where the centripetal force, frictional force, and net force play an important role in determining the motion of the object.

Derive the relation between the angle of banking and design speed

The relationship between the angle of banking and design speed refers to the relationship between the angle at which a road or track is banked (slanted) and the maximum speed at which a vehicle can safely negotiate the bend. This relationship is important in the design of roads and tracks, as it determines the safety of vehicles travelling at high speeds. The relationship can be derived as follows:

  1. Centripetal Force: The force that acts towards the center of a circular path and keeps an object moving in a circular path is known as the centripetal force. The centripetal force can be expressed as:

F = m * a

where “F” is the centripetal force, “m” is the mass of the object, and “a” is the centripetal acceleration.

  1. Normal Force: The normal force is the force exerted by a surface perpendicular to an object in contact with it. In the case of a vehicle negotiating a bend, the normal force acts vertically upwards and is equal in magnitude to the weight of the vehicle.
  2. Frictional Force: The frictional force is the force exerted by a surface on an object in contact with it in the direction opposite to the motion of the object. In the case of a vehicle negotiating a bend, the frictional force acts horizontally and provides the necessary centripetal force to keep the vehicle moving in a circular path.
  3. Relation between Frictional Force and Normal Force: The relationship between the frictional force and the normal force is given by:

F friction = μ * F normal

where “F friction” is the frictional force, “μ” is the coefficient of friction, and “Fnormal” is the normal force.

  1. Relation between Angle of Banking and Frictional Force: The angle of banking is defined as the angle at which a road or track is slanted. The frictional force required to keep a vehicle moving in a circular path depends on the angle of banking and the speed of the vehicle. The relationship can be expressed as:

tan(θ) = μ x F normal / F centripetal

where “θ” is the angle of banking, “μ” is the coefficient of friction, “F normal” is the normal force, and “F centripetal” is the centripetal force.

  1. Relation between Angle of Banking and Design Speed: The design speed is the maximum speed at which a vehicle can safely negotiate a bend. The relationship between the angle of banking and the design speed can be expressed as:

v2 = r * g * tan(θ)

where “v” is the design speed, “r” is the radius of the circular path, “g” is the acceleration due to gravity, and “θ” is the angle of banking.

In conclusion, the relationship between the angle of banking and design speed is important in the design of roads and tracks, as it determines the maximum speed at which a vehicle can safely negotiate a bend. The relationship is derived using the concepts of centripetal force, normal force, frictional force, and the angle of banking, and can be used to design roads and tracks that are safe for vehicles travelling at high speeds.

Recall the concept of Skidding and Overturning on banked road

The concepts of skidding and overturning on banked roads refer to the two potential hazards that vehicles may face when negotiating a bend on a banked road.

  1. Skidding: Skidding occurs when the frictional force between the tires of a vehicle and the road surface is not sufficient to provide the necessary centripetal force to keep the vehicle moving in a circular path. This can cause the vehicle to slide out of control, potentially resulting in an accident. Skidding is more likely to occur when a vehicle is travelling too fast, when the road surface is wet or slippery, or when the angle of banking is not sufficient to provide the necessary centripetal force.
  2. Overturning: Overturning occurs when a vehicle tilts over onto its side or roof while negotiating a bend on a banked road. This can occur if the angle of banking is too steep, if the vehicle is travelling too fast, or if the radius of the bend is too small. When a vehicle overturns, the normal force exerted by the road surface becomes insufficient to keep the vehicle upright, and the centrifugal force causes the vehicle to tip over.

To prevent these hazards, it is important to design roads and tracks with the appropriate angle of banking and radius of curvature for the intended design speed. The angle of banking should be sufficient to provide the necessary centripetal force to keep vehicles moving in a circular path, but not so steep as to cause overturning. The radius of curvature should be large enough to allow vehicles to negotiate the bend without skidding or overturning.

In conclusion, the concepts of skidding and overturning on banked roads are important considerations in the design of roads and tracks. To ensure the safety of vehicles travelling at high speeds, roads and tracks should be designed with the appropriate angle of banking and radius of curvature for the intended design speed.

Describe the Angular Motion

Angular motion refers to the type of motion in which an object rotates around a fixed point, known as the axis of rotation. This type of motion is characterised by an angular displacement, which is the change in the orientation of the object, and an angular velocity, which is the rate of change of the angular displacement.

In angular motion, the object moves in a circular path, with the radius of the circle defined by the distance from the axis of rotation to the object. The magnitude of the angular velocity is determined by the speed of the object along the circular path, and the direction of the angular velocity is perpendicular to the plane of the circular path.

There are two key factors that determine the angular motion of an object: the torque acting on the object and the moment of inertia of the object. Torque is a measure of the force that causes an object to rotate, and is proportional to the magnitude of the force and the distance from the axis of rotation. The moment of inertia of an object is a measure of its resistance to rotation and depends on the distribution of mass within the object.

Angular motion is an important concept in many fields, including physics, engineering, and astronomy. It is used to explain the motion of objects such as gears, pulleys, and spinning tops, as well as the rotation of planets, stars, and galaxies. Understanding angular motion is also important for designing and analysing rotating systems, such as engines, turbines, and generators.

In conclusion, angular motion is a type of motion in which an object rotates around a fixed point, characterised by an angular displacement and an angular velocity. The angular motion of an object is determined by the torque acting on the object and the moment of inertia of the object. Understanding angular motion is important for explaining the motion of rotating objects and for designing and analysing rotating systems.

Derive a relation between angular motion and linear motion

Angular motion and linear motion are two different types of motion that are related to each other. Angular motion refers to the rotation of an object around a fixed point, while linear motion refers to the motion of an object along a straight line. Despite being different types of motion, angular and linear motion are interrelated, and it is possible to describe one in terms of the other.

One of the key ways in which angular and linear motion are related is through the concept of tangential velocity. Tangential velocity is the velocity of a point on an object as it moves in a circular path around an axis of rotation. It is equal to the product of the angular velocity of the object and the radius of the circular path.

Given a specific tangential velocity, we can calculate the linear velocity of an object by dividing the tangential velocity by the radius of the circular path. Conversely, given the linear velocity of an object, we can calculate its tangential velocity by multiplying the linear velocity by the radius of the circular path.

Another way in which angular and linear motion are related is through the concept of centripetal acceleration. Centripetal acceleration is the acceleration that acts towards the center of a circular path and is necessary to keep an object moving in a circular path. It is equal to the square of the tangential velocity divided by the radius of the circular path.

Given the centripetal acceleration of an object, we can calculate its tangential acceleration by dividing the centripetal acceleration by the radius of the circular path. Conversely, given the tangential acceleration of an object, we can calculate its centripetal acceleration by multiplying the tangential acceleration by the radius of the circular path.

In conclusion, angular and linear motion are related to each other through the concepts of tangential velocity and centripetal acceleration. By understanding the relationship between these two types of motion, it is possible to describe one in terms of the other, and to gain a deeper understanding of the motion of objects in both circular and straight-line paths.

Derive the expressions for: i. Uniformly angular velocity ii. Uniformly accelerated rotation

In physics, there are two types of angular motion that are commonly studied: uniformly angular velocity and uniformly accelerated rotation.

i. Uniformly Angular Velocity:

Uniformly angular velocity is a type of angular motion in which an object rotates at a constant rate, with no change in its angular velocity. This means that the object moves through equal angles in equal time intervals.

The expression for the uniformly angular velocity (ω) of an object can be derived using the following formula:

ω = Δθ / Δt

Where Δθ is the change in angular displacement and Δt is the change in time. The angular velocity (ω) has units of radians per second (rad/s).

ii. Uniformly Accelerated Rotation:

Uniformly accelerated rotation is a type of angular motion in which an object rotates at an ever-increasing rate, with a constant angular acceleration (α). This means that the object’s angular velocity is changing at a constant rate.

The expression for the uniformly accelerated rotation of an object can be derived using the following formulae:

α = Δω / Δt

ω = ω0 + αt

θ = θ0 + ω0t + (1/2)αt2

Where α is the angular acceleration, ω0 is the initial angular velocity, ω is the final angular velocity, θ0 is the initial angular displacement, θ is the final angular displacement, t is the time, and Δω and Δt are the changes in angular velocity and time, respectively.

In conclusion, uniformly angular velocity and uniformly accelerated rotation are two important types of angular motion that are studied in physics. The expression for uniformly angular velocity can be derived using the formula Δθ / Δt, while the expression for uniformly accelerated rotation can be derived using the formulae α = Δω / Δt, ω = ω0 + αt, and θ = θ0 + ω0t + (1/2)αt2. Understanding these expressions is important for analyzing and understanding the motion of rotating objects.

Explain the kinetics of rigid body rotation

The kinetics of rigid body rotation refers to the study of the motion and forces involved in the rotation of a rigid body, which is an object that retains its shape and size even under the application of external forces. The field of rigid body rotation is a subfield of classical mechanics, which deals with the motion of objects under the influence of forces.

In the study of rigid body rotation, the key concepts that are considered include:

  1. Angular velocity: This refers to the rate of change of angular displacement of an object, with units of radians per second (rad/s).
  2. Angular acceleration: This refers to the rate of change of angular velocity of an object, with units of radians per second squared (rad/s2).
  3. Moment of Inertia: This refers to a measure of an object’s resistance to rotational motion about a particular axis. It is a scalar quantity and is dependent on the distribution of mass within the object.
  4. Torque: This refers to a force that causes an object to rotate around an axis. Torque is calculated as the product of the force and the lever arm (the perpendicular distance from the axis of rotation to the line of action of the force).
  5. Conservation of angular momentum: This states that the angular momentum of a closed system remains constant unless acted upon by an external torque.
  6. Equations of motion for rigid body rotation: The equations of motion for rigid body rotation can be derived using the laws of classical mechanics. These equations describe the relationship between the forces acting on the object, its moment of inertia, and its angular velocity and acceleration.

In conclusion, the kinetics of rigid body rotation is a branch of classical mechanics that deals with the motion and forces involved in the rotation of rigid bodies. Understanding the key concepts of angular velocity, angular acceleration, moment of inertia, torque, conservation of angular momentum, and the equations of motion for rigid body rotation is important for analyzing and understanding the motion of rotating objects.

Define the following terms related to projectile i. Velocity of projection ii. Angle of projection iii. Trajectory iv. Horizontal range v. Time of flight

Projectile motion refers to the motion of an object that is thrown or projected into the air and then is subject to the influence of gravity. The following terms are related to projectile motion and are important to understand:

  1. Velocity of projection: This refers to the initial velocity of the projectile when it is thrown or projected into the air. It is the vector quantity that determines the initial speed and direction of the projectile.
  2. Angle of projection: This refers to the angle at which the projectile is thrown or projected into the air. It is the angle between the velocity of projection and the horizontal axis.
  3. Trajectory: This refers to the path that the projectile follows as it moves through the air. The trajectory is determined by the initial velocity of projection, the angle of projection, and the effect of gravity.
  4. Horizontal range: This refers to the horizontal distance that the projectile covers from its starting point to its landing point. It is determined by the initial velocity of projection and the angle of projection, and is independent of the effect of gravity.
  5. Time of flight: This refers to the time that the projectile takes to cover the entire distance from its starting point to its landing point. It is determined by the initial velocity of projection, the angle of projection, and the effect of gravity.

In conclusion, understanding the velocity of projection, angle of projection, trajectory, horizontal range, and time of flight is important for analyzing and understanding projectile motion. These terms provide a basis for understanding the motion and behavior of projectiles and are widely used in a variety of applications, such as ballistics, sports, and engineering design.

Describe the motion of body projected horizontally

The motion of a body projected horizontally refers to the motion of an object that is thrown or launched horizontally with a constant velocity. In this type of motion, the initial velocity of the body is purely horizontal and the vertical component of velocity is zero. This means that the only force acting on the body is the gravitational force, which causes it to experience a vertical acceleration due to gravity.

The motion of a body projected horizontally can be analyzed by considering the two components of velocity, the horizontal velocity and the vertical velocity. The horizontal velocity remains constant throughout the motion and does not change, while the vertical velocity changes due to the acceleration due to gravity.

The vertical position of the body can be described by the equation:

y = y + vyt – (1/2)gt2

where y0 is the initial vertical position, v0y is the initial vertical velocity, g is the acceleration due to gravity, and t is time.

The horizontal position of the body can be described by the equation:

x = x + vx t

where x0 is the initial horizontal position and v0x is the initial horizontal velocity.

The time of flight, which is the time taken by the body to reach the ground, can be determined using the following equation:

t = √(2h/g)

where h is the height from which the body was projected and g is the acceleration due to gravity.

In conclusion, the motion of a body projected horizontally is a combination of horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity. Understanding this type of motion is important for analyzing and understanding a variety of real-world situations, such as projectile motion in sports, the motion of objects launched from a cliff, and the motion of satellites in space.

Describe the inclined projection on level ground

Inclined projection on level ground refers to the motion of a body that is projected upwards at an angle to the horizontal. In this type of motion, the body experiences both horizontal and vertical motion.

The velocity of the body at the moment of projection can be divided into two components: the horizontal velocity (v0x) and the vertical velocity (v0y). The horizontal velocity is constant and remains the same throughout the motion, while the vertical velocity changes due to the acceleration due to gravity (g).

The motion of a body projected upwards at an angle can be analyzed by considering the equations of motion in the x and y direction. The horizontal position of the body can be described by the equation:

x = x + vx t

where x0 is the initial horizontal position and v0x is the initial horizontal velocity.

The vertical position of the body can be described by the equation:

y = y + vyt – (1/2)gt2

where y0 is the initial vertical position, v0y is the initial vertical velocity, g is the acceleration due to gravity, and t is time.

The time of flight, which is the time taken by the body to reach the maximum height, can be determined using the following equation:

t = vy/g

The range of the body, which is the horizontal distance covered by the body, can be determined using the following equation:

R = (v2 sin(2θ))/g

where θ is the angle of projection.

In conclusion, the motion of a body projected upwards at an angle on level ground is a combination of horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity. Understanding this type of motion is important for analyzing and understanding a variety of real-world situations, such as the trajectory of a thrown ball or the flight path of a missile.

Describe the projection at different levels with the point of projection and point of strike

Projection at different levels refers to the motion of a body that is projected from one level to another level, either horizontally or at an angle. The point of projection is the initial position of the body at the moment of projection, while the point of strike is the final position of the body at the end of its motion.

When a body is projected from a higher level to a lower level, it undergoes free fall motion. The body experiences a vertical acceleration due to gravity, and its velocity in the vertical direction increases as it falls. The horizontal velocity of the body remains constant throughout the motion.

When a body is projected from a lower level to a higher level, such as a ball being thrown upwards, it experiences a combination of vertical and horizontal motion. The vertical velocity of the body decreases due to the acceleration due to gravity, while the horizontal velocity remains constant.

In both cases, the motion of the body can be analysed using the equations of motion in the x and y direction. The vertical position of the body can be described by the equation:

y = y + vy t – (1/2)gt2

where y0 is the initial vertical position, v0y is the initial vertical velocity, g is the acceleration due to gravity, and t is time.

The horizontal position of the body can be described by the equation:

x = x + vx t

where x0 is the initial horizontal position and v0x is the initial horizontal velocity.

The time of flight, which is the time taken by the body to reach the point of strike, can be determined using the vertical velocity and acceleration due to gravity. The range of the body, which is the horizontal distance covered by the body, can be determined using the horizontal velocity and time of flight.

In conclusion, the projection of a body at different levels can be analyzed using the equations of motion in the x and y direction, and the final position of the body (point of strike) can be determined based on its initial position (point of projection), initial velocity, and acceleration due to gravity. Understanding this type of motion is important for analyzing and understanding a variety of real-world situations, such as the motion of a thrown ball or the flight path of a projectile.