In preparation for this problem, review Conceptual Example 7. From the top of a cliff, a person uses a slingshot to fire a pebble straight downward, which is the negative direction. The initial speed of the pebble is 6.46 m/s. (a) What is the acceleration (magnitude and direction) of the pebble during the downward motion? (b) After 0.667 s, how far beneath the cliff top is the pebble?
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To answer these questions, let's break them down step by step.
(a) What is the acceleration (magnitude and direction) of the pebble during the downward motion?
To find the acceleration of the pebble, we can use the formula:
acceleration (a) = change in velocity (Δv) / time (t)
In this case, the pebble is moving straight downward, so the magnitude of its acceleration can be calculated by taking the difference in velocity between the final velocity and the initial velocity. Since the initial velocity is 6.46 m/s and the pebble is moving downward, the final velocity would be smaller. Therefore, we can say the final velocity is 0 m/s.
So, the acceleration (a) can be calculated as:
a = (0 - 6.46) m/s / t
Since the pebble is moving freely under gravity, we know that acceleration due to gravity (g) is acting on it. The magnitude of acceleration due to gravity on Earth is 9.8 m/s^2, and its direction is downward. So, we can write the equation as:
a = -g
Therefore, the magnitude of acceleration of the pebble during downward motion is 9.8 m/s^2, and its direction is downward.
(b) After 0.667 s, how far beneath the cliff top is the pebble?
To find the distance the pebble has fallen after 0.667 s, we can use the formula:
distance (y) = initial velocity (v₀) * time (t) + 0.5 * acceleration (a) * time (t)^2
Since the pebble is moving straight downward, the initial velocity (v₀) is 6.46 m/s and the acceleration (a) is 9.8 m/s^2 (as we calculated earlier).
Plugging in the values:
y = (6.46 m/s) * (0.667 s) + 0.5 * (9.8 m/s^2) * (0.667 s)^2
Simplifying the equation, we get:
y ≈ -2.18 m
Therefore, after 0.667 s, the pebble is approximately 2.18 meters beneath the cliff top.