A ball is projected horizontally from the top of a 31.1 m high cliff. The ball lands on the ground after travelling a horizontal range of 29.1 m.Ignoring air resistance, what was the initial velocity of the ball?

Question

BlackTom AI · Accepted Answer

Horizontal projectile motion separates into independent horizontal and vertical components; the horizontal speed is constant while the vertical motion is free fall under gravity. Use vertical motion to find the time of flight, then use horizontal distance to get the initial horizontal velocity.

Relevant formulas and concepts:
- Vertical displacement for free fall: y = (1/2) g t^2 (initial vertical velocity = 0).
- Horizontal speed (constant): v_x = horizontal range / time.

Step-by-step calculation:
1. Known: vertical drop y = 31.1 m, g = 9.81 m/s^2.
   Use y = (1/2) g t^2 → t = sqrt(2y/g).
2. Substitute numbers: t = sqrt(2 × 31.1 m / 9.81 m/s^2).
   Compute numerator: 2 × 31.1 = 62.2.
   Division: 62.2 / 9.81 ≈ 6.341.
   Square root: t ≈ sqrt(6.341) ≈ 2.518 s.
3. Horizontal range x = 29.1 m, so v_x = x / t = 29.1 m / 2.518 s.
   Division: v_x ≈ 11.56 m/s.

Why this is correct:
- The calculation uses independent-motion principles correctly: vertical fall determines the time, horizontal displacement divided by that time gives the constant horizontal speed. Numerical substitution and arithmetic yield v_x ≈ 11.56 m/s (rounded appropriately).

Why the other choices are incorrect:
- a. 23.11 m/s is about double the correct value — it would imply a flight time roughly half the true time, inconsistent with the vertical fall time.
- c. 5.78 m/s is about half the correct value — it would require a flight time about twice the true time, which contradicts the computed free-fall time.
- d. 16.34 m/s does not match the horizontal range divided by the correct time and so is inconsistent with the vertical fall duration.

Conclusion: The initial velocity (horizontal) is 11.56 m/s, matching the provided correct answer (option b).

A ball is projected horizontally from the top of a 31.1 m high cliff. The ball lands on the ground after travelling a horizontal range of 29.1 m.Ignoring air resistance, what was the initial velocity of the ball?单项选择题

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