a soccer player kicks a rock horizontally off a 40.0-m-high cliff into a pool of water. if the player hears the sound of the splash 3.00 s later, what was the initial speed given to the rock? assume the speed of sound in air to be 343m/s.
To find the initial speed given to the rock, we can use the fact that the time it takes for the sound to reach the player is equal to the time it takes for the rock to fall to the water.
First, let's determine the time it takes for the rock to fall to the water using the equation of motion for free-falling objects:
h = (1/2) * g * t^2
where:
h = height (40.0 m)
g = acceleration due to gravity (9.8 m/s^2)
t = time
Rearranging the equation to solve for time:
t = sqrt((2h) / g)
t = sqrt((2 * 40.0 m) / 9.8 m/s^2)
t ≈ 2.02 s (rounded to two decimal places)
Since the time it takes for the sound to reach the player is 3.00 s, we can subtract the time it takes for the rock to fall from this total:
time taken by sound - time taken by rock = 3.00 s - 2.02 s = 0.98 s
Now, let's determine the distance traveled by the sound using the equation:
distance = speed × time
where:
distance = 40.0 m (height of the cliff)
time = 0.98 s (time taken for the sound)
speed = distance / time
speed = 40.0 m / 0.98 s ≈ 40.82 m/s (rounded to two decimal places)
Therefore, the initial speed given to the rock is approximately 40.82 m/s.
To find the initial speed given to the rock, we can use the fact that the time it takes for the sound to reach the player is equal to the time it takes for the rock to fall and hit the water.
Let's breakdown the problem into different parts:
1. Determine the time it takes for the rock to fall from the cliff to the water surface:
The height of the cliff is given as 40.0 m. We can use the equation for free fall motion:
h = 1/2 * g * t^2, where h is the height, g is the acceleration due to gravity, and t is the time.
Rearranging the equation to solve for t:
t = sqrt(2h / g), where sqrt stands for square root.
t = sqrt(2 * 40.0 / 9.8)
t ≈ 2.03 s
2. Determine the time it takes for the sound to reach the player:
The time taken for sound to travel can be found using the distance formula:
distance = speed * time
Given that the speed of sound in air is 343 m/s and the time is 3.00 s:
343 * 3 = 1029 m
3. Equate the times:
Since the time for the rock to fall is approximately 2.03 s and the time for the sound to reach the player is 3.00 s, we can set up the equation:
2.03 + x = 3.00
Solving for x (the time it takes for the sound to reach the player after the rock splash):
x ≈ 0.97 s
4. Calculate the initial speed given to the rock:
Using the equation for horizontal motion:
distance = speed * time
The distance traveled by the sound in 0.97 s is 1029 m. Therefore:
distance = speed * time
1029 = speed * 0.97
Solving for the initial speed:
speed ≈ 1061.86 m/s
Hence, the initial speed given to the rock is approximately 1061.86 m/s.
Neglecting air friction, the time it will take the rock to hit the water is
sqrt(2H/g) = 2.857 seconds. That means the sound of the splash takes 3 - 2.857 = 0.143 seconds to be heard, after hitting water. Sound travelled 49.05 m in that time. Since the cliff is 40 m high, the horizontal distance travelled by the rock was (from the Pythagorean theorem)
sqrt[(49.05)^2 -(40)^2] = 28.4 m
The rock's initial speed (and constant horizontal speed) was 28.4 m/2.857 s = 9.84 m/s