Derive the equation of motion for a projectile moving under the influence of gravity

A projectile is an object that is launched into the air and moves under the influence of gravity. The motion of a projectile is a classic example of a two-dimensional motion problem. 

The equation of motion for a projectile moving under the influence of gravity is derived by considering the forces acting on the projectile and applying Newton's Second Law.

Derive the equation of motion for a projectile moving under the influence of gravity-Consider a projectile of mass m that is launched into the air with an initial velocity v0 at an angle θ above the horizontal. 

The projectile moves under the influence of gravity, which acts downward with a force of mg, where g is the acceleration due to gravity. The motion of the projectile can be described by its position vector r(t), which gives the position of the projectile at time t.

The forces acting on the projectile are the gravitational force mg and the air resistance force, which we assume to be negligible. The gravitational force acts downward and can be resolved into two components: one parallel to the horizontal direction and one parallel to the vertical direction. 

Derive the equation of motion for a projectile moving under the influence of gravity-The component parallel to the horizontal direction does not affect the motion of the projectile in the vertical direction and can be ignored. The component parallel to the vertical direction is given by:

• Fy = -mg

where the negative sign indicates that the force is acting downward.

The equation of motion for the projectile can be derived by applying Newton's Second Law, which states that the net force acting on an object is equal to its mass times its acceleration. In the case of the projectile, the net force acting on it is the gravitational force, and its acceleration is given by the second derivative of its position vector with respect to time:

• Fnet = ma = m(d^2r/dt^2)

where a is the acceleration of the projectile.

Substituting the gravitational force for the net force, we have:

• -mg = m(d^2r/dt^2)

Dividing both sides by m, we obtain:

• -g = d^2r/dt^2

This is the equation of motion for the projectile under the influence of gravity. It is a second-order differential equation that describes the vertical motion of the projectile. The solution to this equation gives the vertical position of the projectile as a function of time.

To solve this equation, we need to specify the initial conditions of the projectile, namely its initial position and velocity. Let the initial position of the projectile be r0 = (x0, y0), where x0 is the horizontal distance from the origin and y0 is the initial height above the ground. Let the initial velocity of the projectile be v0 = (v0x, v0y), where v0x is the horizontal component of the velocity and v0y is the vertical component of the velocity.

Also Read-

• Explain The Concept Of Elasticity And The Stress-Strain Relationship For Linearly Elastic Materials
• Derive The Equations Of Motion For A Rigid Body Undergoing Planar Motion
• Explain The Principle Of Virtual Work And How It Can Be Used To Solve Mechanical Problems
• Describe The Concept Of Work-Energy Principle And Give An Example Of Its Application

The initial conditions can be used to determine the constants of integration in the solution to the equation of motion. Integrating the equation with respect to time gives:

• -dy/dt = gt + vy0

Integrating again with respect to time gives:

• y = y0 + vy0t - 1/2gt^2

Derive the equation of motion for a projectile moving under the influence of gravity-This is the equation of motion for the vertical position of the projectile as a function of time. It gives the height of the projectile above the ground at any time t.

Similarly, we can derive the equation of motion for the horizontal position of the projectile. The horizontal motion of the projectile is not affected by gravity, so the net force acting on it is zero. The equation of motion for the horizontal position of the projectile is:

• dx/dt = vx0

where vx0 is the horizontal component of the initial velocity.

The equations of motion for the vertical and horizontal positions of the projectile can be combined to give the trajectory of the projectile. The trajectory is given by the parametric equations:

• x = x0 + vx0t
• y = y0 + vy0t - 1/2gt^2

Derive the equation of motion for a projectile moving under the influence of gravity-These equations give the horizontal and vertical positions of the projectile as functions of time. The trajectory of the projectile is the path traced out by these positions as the projectile moves through the air.

Conclusion

The equation of motion for a projectile moving under the influence of gravity is derived by considering the forces acting on the projectile and applying Newton's Second Law. 

Derive the equation of motion for a projectile moving under the influence of gravity-The equation of motion is a second-order differential equation that describes the vertical motion of the projectile. The solution to this equation gives the vertical position of the projectile as a function of time. 

Derive the equation of motion for a projectile moving under the influence of gravity-The horizontal motion of the projectile is not affected by gravity and can be described by a simple linear equation. The trajectory of the projectile is given by the parametric equations that combine the equations of motion for the vertical and horizontal positions of the projectile.

FAQ.

Q. What is the equation of motion for a projectile under gravity?

Ans. The equation of motion for a projectile moving under the influence of gravity is given by: -g = d^2r/dt^2, where g is the acceleration due to gravity and r(t) is the position vector of the projectile at time t.

Q. What does the equation of motion describe?

Ans. The equation of motion describes the vertical motion of the projectile, specifically the second derivative of its position vector with respect to time.

Q. How is the equation of motion derived?

Ans. The equation of motion is derived by considering the forces acting on the projectile, namely gravity, and applying Newton's Second Law.

Q. What are the initial conditions needed to solve the equation of motion?

Ans. The initial conditions needed to solve the equation of motion are the initial position and velocity of the projectile, which can be used to determine the constants of integration in the solution.

Q. How accurate is the equation of motion in real-world situations?

Ans. The equation of motion assumes certain ideal conditions, such as negligible air resistance and a constant acceleration due to gravity. In real-world situations, these conditions may not hold, and the motion of the projectile may be more complex.

Q. Can the equation of motion be used for non-vertical motion?

Ans. The equation of motion derived is specifically for the vertical motion of the projectile. For non-vertical motion, additional equations and considerations may be needed to describe the motion accurately.

• Tweet
• Share
• Share
• Share