Projectiles

Projectile motion refers to the motion of a body that travels freely in space under the influence of gravity and air resistance. When a ball is kicked into the air, it travels through space in a plane, combining both upward and horizontal motion.

The path through which a projectile travels is called the trajectory.

Examples of Projectile Motion

• In sports:
  • Throwing a discus
  • Throwing a javelin
• In warfare:
  • Firing a catapult
  • Shooting arrows with a bow
  • Launching missiles
• Miscellaneous:
  • Throwing stones

Projectile Motion at an Angle (θ) to the Horizontal

When a body is projected at an angle, its motion can be split into two components:

I. Horizontal Motion

In the horizontal motion, the body moves with constant velocity, meaning the horizontal acceleration is zero. This also implies that the initial and final horizontal velocities are equal.

Mathematically:

• ax = 0
• Vx = Ux

Where U is the initial velocity with which the body was projected. Resolving U into its vertical and horizontal components, we get:

• U_x = U cos 𝜃
• U_y = U sin 𝜃

II. Vertical Motion

The vertical motion is an example of uniformly accelerated motion. The equations of uniform motion are still valid for it.

During the upward motion:

• Vertical acceleration: ay = -g (where g is the acceleration due to gravity)
• Initial vertical velocity: Uy = U sin 𝜃
• At the maximum height, the body is temporarily at rest, so Vy = 0

Projectile Equations

Height

This is the highest maximum vertical displacement reached by the projected object

$$ H = \frac{U^2\sin^2\theta}{2g}$$

Distance Covered

Vertically

$$ U_y = ut\sin\theta - \frac{1}{2}gt^2 $$

Horizontally

$$ U_x = ut\cos\theta $$

Range

The horizontal range R of the projectile is the horizontal distance it has traveled when it returns to its initial height.

$$ R = \frac{U^2\sin2\theta}{g}$$

Max Range

It is the longest distance covered by the object during projectile motion. When the angle of projection is 45°, the maximum range is obtained.

Time Taken

Time taken to reach Max height

$$ t = \frac{U\sin\theta}{g}$$

Time of Flight

$$ T = \frac{2U\sin\theta}{g}$$

Velocity

Vertically

$$ U_y = u\sin\theta - gt $$

Horizontally

The velocity is U/stays the same.