In an experiment two identical rocks are simultaneously thrown from the edge of a cliff a distance h0 above the ground. Rock A is thrown vertically upward with speed v0 and rock B is thrown vertically downward with speed v0. Rock A and rock B strike the ground at times tA and tB, respectively. Consider the positive vertical direction to be upward.
(a) On the axes given below, sketch and label graphs of the velocity as a function of time for rock A and rock B. Label the time tB. Times tA and 1/2tA are given on the graph.
The Correct Answer and Explanation is:
Correct Answer (Velocity vs. Time Graph Explanation)
To sketch the velocity vs. time graph for both Rock A and Rock B:
- Use a vertical axis for velocity (v) and a horizontal axis for time (t).
- Let upward be positive, so gravity will cause a negative acceleration of −g (−9.8 m/s²).
- Both rocks are thrown from the same height and at the same speed (v₀), one upward and the other downward.
Key Events and Labels:
- Rock A:
- Starts at +v₀ (positive, upward velocity).
- Slows down due to gravity, reaches 0 velocity at the top of its path at t = ½tA, then starts falling downward.
- Continues to accelerate downward and becomes negative in velocity.
- Hits the ground at t = tA with a larger magnitude of velocity than it started with.
- Rock B:
- Starts at −v₀ (negative, downward velocity).
- Continues to accelerate downward at −g.
- Hits the ground sooner than rock A, at t = tB < tA.
On the graph:
- Rock A: A straight line starting at +v₀ on the velocity axis, decreasing linearly due to gravity (slope = −g), reaching 0 at t = ½tA, and then becoming negative, ending at a more negative velocity at t = tA.
- Rock B: A straight line starting at −v₀, with the same slope (−g) as Rock A, but since it doesn’t go upward first, it hits the ground at t = tB, which is earlier than tA.
Both lines should be straight with equal slopes because gravity affects both identically.
Explanation (300+ words)
This experiment is a classic physics problem involving one-dimensional motion under constant acceleration (gravity). The key insight is that both rocks experience the same acceleration due to gravity (g ≈ 9.8 m/s² downward), but they are thrown in opposite directions initially — one upward (Rock A) and one downward (Rock B).
For Rock A, it is thrown upward with an initial velocity of +v₀. Since gravity acts downward, it slows the rock until its vertical velocity becomes 0 m/s at the peak of its path. This point occurs at t = ½tA. Afterward, the rock begins falling back down, gaining speed in the negative (downward) direction due to gravity. By the time it hits the ground at t = tA, its velocity is negative and greater in magnitude than +v₀ due to the total distance it has traveled (up and then down past the starting point).
For Rock B, it is thrown downward with an initial velocity of −v₀. Gravity accelerates it further downward. Since it does not need to first reverse direction, it hits the ground before Rock A, at t = tB < tA. Its velocity continues becoming more negative due to gravity’s constant downward pull.
The velocity vs. time graph for both rocks are straight lines with the same slope (−g). The difference lies in their starting points and duration. Rock A’s graph starts at +v₀ and passes through zero at t = ½tA, while Rock B starts at −v₀ and ends sooner. The intersection at t = ½tA on Rock A’s graph represents the moment it changes direction.
This graph visually demonstrates how symmetric vertical motion is under constant acceleration and emphasizes the importance of direction in velocity.