Section 1.4: Projection from a Height

In horizontal projection from a height, an object is launched horizontally from an elevated position. The vertical motion is free fall, while the horizontal motion has constant velocity.

Assumptions:
  • Initial vertical velocity \(v_{0y} = 0\)
  • Horizontal velocity \(v_x = v_0\) constant
  • Acceleration due to gravity \(g = 9.8\text{ m/s²}\) downward
  • Neglect air resistance

Equations of Motion

  • Vertical: \(y = h - \frac{1}{2} g t^2\) (height decreases with time)
  • Time to hit ground: \(t = \sqrt{\frac{2h}{g}}\)
  • Horizontal: \(x = v_x t = v_0 \sqrt{\frac{2h}{g}}\)
  • Horizontal displacement (range) depends on height and horizontal velocity only

Example 1

A ball is projected horizontally from a height of 20 m with speed 10 m/s. Find time to hit the ground and horizontal distance traveled.

Time: \(t = \sqrt{2h/g} = \sqrt{2*20/9.8} \approx 2.02 \text{ s}\)

Horizontal distance: \(x = v_x t = 10 * 2.02 \approx 20.2 \text{ m}\)

Example 2

A stone is thrown horizontally from a cliff 45 m high with speed 15 m/s. Calculate the time to reach the ground and the horizontal range.

Time: \(t = \sqrt{2*45/9.8} \approx 3.03 s\)

Horizontal range: \(x = 15 * 3.03 \approx 45.5 m\)

Practice Problems

  1. A ball is projected horizontally from a 25 m high tower with speed 12 m/s. Find time to hit the ground and horizontal distance.
  2. A stone is thrown horizontally from 50 m height with speed 8 m/s. Determine time of flight and horizontal displacement.
  3. An object is launched horizontally from a 10 m high balcony at 5 m/s. How far from the base will it land?
  4. A cannon fires a shell horizontally from a height of 30 m with speed 20 m/s. Find time of flight and range.
  5. A ball is projected horizontally from a 40 m tall building at 15 m/s. Calculate the time it takes to reach the ground and the horizontal distance covered.
  6. An object is thrown horizontally from 60 m height at 18 m/s. Find landing time and horizontal displacement.
  7. A stone is projected from 80 m cliff at 25 m/s. Determine how far from the base it lands.
  8. A ball rolls off a table 1.5 m high at 3 m/s. Find how long it takes to reach the floor and horizontal distance traveled.
  9. A rock is thrown horizontally from a 12 m height with speed 6 m/s. Determine time to hit ground and horizontal range.
  10. An object is projected horizontally from 100 m height at 10 m/s. Find time and horizontal displacement.
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