A stone is dropped at rest from the top of a cliff. It is observed to hit the ground
5.78 s later. How high is the cliff?
The correct answer and explanation is :
To determine the height of the cliff, we can use the equations of motion for an object in free fall. When a stone is dropped from rest, its initial velocity is zero, and it is affected only by gravity. The acceleration due to gravity is ( g = 9.8 \, \text{m/s}^2 ).
We can use the following kinematic equation to find the height (h) the stone falls:
[
h = v_0 t + \frac{1}{2} g t^2
]
Where:
- ( h ) is the height of the cliff (the distance the stone falls),
- ( v_0 ) is the initial velocity of the stone, which is 0 since the stone is dropped from rest,
- ( g ) is the acceleration due to gravity (9.8 m/s²),
- ( t ) is the time the stone takes to fall, which is given as 5.78 seconds.
Since ( v_0 = 0 ), the equation simplifies to:
[
h = \frac{1}{2} g t^2
]
Now, we can substitute the known values into the equation:
[
h = \frac{1}{2} \times 9.8 \, \text{m/s}^2 \times (5.78 \, \text{s})^2
]
First, square the time:
[
(5.78)^2 = 33.49 \, \text{s}^2
]
Now, calculate the height:
[
h = \frac{1}{2} \times 9.8 \times 33.49
]
[
h = 4.9 \times 33.49
]
[
h = 164.1 \, \text{m}
]
So, the height of the cliff is approximately 164.1 meters.
Explanation:
The motion of the stone is influenced only by gravity, and since it’s dropped (not thrown), the initial velocity is zero. The acceleration due to gravity is constant, so we can use the basic kinematic equation for an object in free fall to find the distance it falls during the time it is in the air. In this case, we found that after 5.78 seconds, the stone falls a distance of about 164.1 meters. This is the height of the cliff from which it was dropped.