Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air and is influenced only by the force of gravity (assuming negligible air resistance). The path followed is a curved parabola.

General Features

• The motion is two-dimensional: horizontal and vertical components.
• The horizontal component has constant velocity (\( a_x = 0 \)).
• The vertical component has uniformly accelerated motion (\( a_y = -g \)).
• The trajectory is parabolic.

Initial Velocity Components

Let the projectile be launched with speed \( u \) at an angle \( \theta \) from the horizontal:

• Horizontal velocity: \( u_x = u\cos\theta \)
• Vertical velocity: \( u_y = u\sin\theta \)

Case 1: Ground-to-Ground Projectile Motion

In this case, the object lands back on the same horizontal level it was projected from. The motion is symmetric.

• Time to reach maximum height: \( t_{\text{up}} = \dfrac{u\sin\theta}{g} \)
• Total time of flight: \( T = \dfrac{2u\sin\theta}{g} \)
• Maximum height: \( H = \dfrac{u^2\sin^2\theta}{2g} \)
• Horizontal range: \( R = \dfrac{u^2\sin(2\theta)}{g} \)

Case 2: Projectile Motion from a Height

In this case, the projectile is launched from an elevated position or lands at a different height from the starting point.

• The total time is found by solving: \( y = u\sin\theta \cdot t – \dfrac{1}{2}gt^2 + h_0 \)
• The range depends on time of flight: \( R = u\cos\theta \cdot t \)
• Maximum height is still given by: \( H = \dfrac{(u\sin\theta)^2}{2g} \) , if we launch a projectile from some initial height h then this “h” will be add in maximum height.