A soccer player kicks a rock horizontally off a 41.7 m high cliff into a pool of water. If the player hears the sound of the splash 3.08 s later, what was the initial speed given to the rock? Assume the speed of sound in air to be 343 m/s.
Use the height of the cliff to predict the time to fall, t1.
t1 = sqrt (2H/g)= 2.917 s
The delay time to hear the splash
t2 = 3.08 - 2.917 s,
tells you how far away it hits the water, S = t2*343 = 55.9 m
which is a combination of vertical and horizontal displacement, H and X. Use the Pythagorean theorem
S^2 = H^2 + X^2
to get the horizontal displacement when the ball hits the water:
X = sqrt[(55.9)^2 - (41.7)^2] = 37.2 m
The initial speed of the rock, which was horizontal, equals X divided by the fall time, t1.
To solve this problem, we can use the fact that the time taken for the sound to reach the player's ears is equal to the time taken for the rock to fall from the cliff to the pool of water.
First, let's find the time taken for the rock to fall from the cliff to the pool of water:
Using the equation of motion for free fall:
h = (1/2) * g * t^2
Where:
h = height of the cliff = 41.7 m
g = acceleration due to gravity = 9.8 m/s^2
t = time taken for the rock to fall
Rearranging the equation, we get:
t^2 = (2h) / g
t^2 = (2 * 41.7) / 9.8
t^2 = (83.4) / 9.8
t^2 ≈ 8.51
Taking the square root of both sides:
t ≈ √8.51
t ≈ 2.92 s (rounded to two decimal places)
Now, we know that the total time taken for the sound to reach the player's ears is 3.08 s. Let's denote the time taken for the sound as t_sound.
So, t_sound = 3.08 s
Now, we can find the time taken for only the rock to fall by subtracting the time taken for the sound from the total time:
t_rock = t_total - t_sound
t_rock = 2.92 s - 3.08 s
t_rock ≈ -0.16 s
We have a negative value for t_rock, which means something is not right. Upon further inspection, we realize that it is not possible for the sound to be heard after the rock hits the water. This condition violates the assumption that the speed of sound in air is greater than the speed of the rock.
Therefore, we cannot find the initial speed given to the rock using the given information.