A rock is thrown vertically upward from the edge of a cliff. The rock reaches a maximum height of 20.6 m above the top of the cliff before falling to the base of the cliff, landing 6.44 s after it was thrown. How high is the cliff?
To determine the height of the cliff, we need to consider the motion of the rock and use the kinematic equations.
Let's break down the problem and gather the given information:
- The rock reaches a maximum height of 20.6 m above the top of the cliff.
- The total time it takes for the rock to reach the maximum height and fall back down to the base of the cliff is 6.44 seconds.
First, we need to find the time it takes for the rock to reach its maximum height. Since the motion is vertical and the only force acting on the rock is gravity, we can use the equation for the time it takes to reach the maximum height:
t = (v_f - v_i) / g
Where:
- t is the time taken to reach the maximum height
- v_f is the final velocity (which is 0 since the rock reaches its peak and starts falling down)
- v_i is the initial velocity
- g is the acceleration due to gravity (-9.8 m/s^2)
Using this equation, we can rearrange it to solve for v_i:
v_i = v_f - g * t
Substituting the given values:
v_i = 0 - (-9.8 m/s^2) * t
Since we want to find the height of the cliff, we need to determine the time it takes for the rock to reach the maximum height. Let's substitute the given total time:
v_i = -9.8 m/s^2 * 6.44 s
Calculating v_i:
v_i = -63.112 m/s
Now that we know the initial velocity, we can use it to calculate the height of the cliff. We can use the equation for the height reached by an object in free fall:
h = v_i * t + 0.5 * g * t^2
Substituting the values:
h = -63.112 m/s * 6.44 s + 0.5 * (-9.8 m/s^2) * (6.44 s)^2
Calculating the height:
h ≈ -204.25 m
This value is negative because we chose the upward direction as positive. Therefore, the height of the cliff is the absolute value of the result:
Height of the cliff ≈ 204.25 m
Therefore, the height of the cliff is approximately 204.25 meters.