A rock is dropped from a 154-m-high cliff. How long does it take to fall (a) the first 77.0 m and (b) the second 77.0 m?
d = Vo*t + 0.5gt^2 = 77,
0 + 4.9t^2 = 77,
t^2 = 15.7,
t = 3.96s. = Time to fall 1st 77m.
d = 0 + 4.9t^2 = 154m,
4.9t^2 = 154,
t^2 = 31.4,
t = 5.6s. = Time to fall to ground.
T = 5.6 - 3.96 = 1.65s. = Time to fall
2nd 77m.
To solve this problem, we can use the equation for the time it takes for an object to fall freely from a certain height:
t = sqrt(2h/g)
where:
t = time (in seconds)
h = height (in meters)
g = acceleration due to gravity (approximately 9.8 m/s^2)
(a) First 77.0 m:
Using the equation, we can substitute the given values:
t = sqrt(2(77.0)/9.8)
t = sqrt(154/9.8)
t = sqrt(15.71)
t ≈ 3.96 seconds
Therefore, it takes approximately 3.96 seconds for the rock to fall the first 77.0 meters.
(b) Second 77.0 m:
The total time it takes for the rock to fall from the top of the cliff (154 m) is the same as the time it takes to fall the second 77.0 m. This is because the acceleration due to gravity is constant.
Therefore, it also takes approximately 3.96 seconds for the rock to fall the second 77.0 meters.
To find the time it takes for a rock to fall, we can use the kinematic equation for vertical motion under constant acceleration:
h = 1/2 * g * t^2
Where:
h = height
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Let's solve part (a) first:
We know that the total height is 154 m, and we want to find the time it takes for the rock to fall the first 77.0 m.
Using the kinematic equation, we can rearrange it to solve for time:
t = sqrt((2 * h) / g)
Plugging in the given values:
h = 77.0 m
g = 9.8 m/s^2
t = sqrt((2 * 77.0) / 9.8)
t = sqrt(154 / 9.8)
t ≈ 4 seconds (rounded to the nearest whole number)
So, it takes approximately 4 seconds for the rock to fall the first 77.0 m.
Now, let's solve part (b):
For the second 77.0 m, we need to find the time difference between the height of 154 m and the first 77.0 m.
Since the time for the first 77.0 m was 4 seconds, we subtract that from the total time.
So, for the second 77.0 m, it takes approximately 4 - 4 = 0 seconds.
Therefore, it takes 0 seconds for the rock to fall the second 77.0 m.