Section 1.3: Projectile Motion

Projectile motion describes the motion of an object launched into the air, moving under the influence of gravity. It is a combination of horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity.

Assumptions:
  • Acceleration due to gravity \(g = 9.8 \text{ m/s²}\) downward
  • Neglect air resistance
  • Horizontal velocity is constant
  • Vertical motion obeys kinematic equations with acceleration \(a_y = -g\)

Components of Motion

  • Horizontal: \(x = x_0 + v_{0x} t\) (constant velocity)
  • Vertical: \(y = y_0 + v_{0y} t - \frac{1}{2} g t^2\)
  • Vertical velocity: \(v_y = v_{0y} - g t\)

The motion is independent in x and y directions, except for sharing the same time.

Key Quantities

  • Time of flight: \(T = \frac{2 v_0 \sin\theta}{g}\)
  • Maximum height: \(H = \frac{(v_0 \sin\theta)^2}{2g}\)
  • Range: \(R = \frac{v_0^2 \sin 2\theta}{g}\)
  • \(v_0\) = initial speed, \(\theta\) = launch angle

Example 1

A ball is thrown with speed 20 m/s at an angle 30° above horizontal. Find time of flight, maximum height, and horizontal range.

Time: \(T = \frac{2 v_0 \sin\theta}{g} = \frac{2*20*\sin30°}{9.8} \approx 2.04 s\)

Max height: \(H = \frac{(v_0 \sin\theta)^2}{2g} = \frac{(20*0.5)^2}{2*9.8} \approx 10.2 m\)

Range: \(R = \frac{v_0^2 \sin2\theta}{g} = \frac{400 * \sin60°}{9.8} \approx 35.3 m\)

Example 2

A cannonball is fired horizontally from a height of 50 m with speed 40 m/s. Find time to hit ground and horizontal distance traveled.

Vertical motion: \(y = y_0 - \frac{1}{2} g t^2 \Rightarrow t = \sqrt{\frac{2*50}{9.8}} \approx 3.19 s\)

Horizontal motion: \(x = v_x t = 40*3.19 \approx 127.6 m\)

Practice Problems

  1. A ball is thrown at 15 m/s at 45° above horizontal. Find maximum height, time of flight, and range.
  2. A projectile is launched with 25 m/s at 60°. Calculate time in air and horizontal distance covered.
  3. A stone is thrown horizontally from 20 m height at 10 m/s. Find time to hit ground and distance traveled.
  4. A ball is thrown at 30 m/s at 30° above horizontal. Find maximum height and range.
  5. A projectile is launched at 18 m/s at 50°. Determine horizontal and vertical components of velocity at t = 1 s.
  6. A cannonball is fired horizontally from a cliff 80 m high at 20 m/s. Find where it lands.
  7. A ball is thrown at 12 m/s at 60°. Find total time of flight and horizontal displacement.
  8. A projectile is launched at 10 m/s at 45°. Find maximum height and range.
  9. A stone is thrown horizontally from 15 m height with speed 8 m/s. Find time to hit the ground and horizontal distance.
  10. A projectile is launched at 20 m/s at 37°. Find time of flight, range, and max height.
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