A gull is flying horizontally 8.17 m above the ground at 6.35 m/s. The bird is carrying a clam in its beak and plans to crack the clamshell by dropping it on some rocks below. Ignoring air resistance, what is the horizontal distance to the rocks at the moment that the gull should let go of the clam?
With what speed relative to the rocks does the clam smash into the rocks?
and with what speed relative to the gull does the clam smash into the rocks?
To find the horizontal distance to the rocks when the gull should let go of the clam, we need to determine the time it takes for the clam to reach the ground.
First, let's calculate the time it takes for the clam to fall to the ground using the formula:
d = (1/2) * g * t^2
Where:
d = vertical distance (8.17 m)
g = acceleration due to gravity (9.8 m/s^2)
t = time
Plugging in the values, we get:
8.17 = (1/2) * 9.8 * t^2
Rearranging the equation to solve for t, we get:
t^2 = (2 * 8.17) / 9.8
t^2 = 1.6776
t ≈ 1.296 s
Now, let's find the horizontal distance traveled by the clam during this time. Since the gull is flying horizontally, the horizontal distance can be calculated using the formula:
d = v * t
Where:
v = horizontal velocity (6.35 m/s)
t = time (1.296 s)
Plugging in the values, we get:
d = 6.35 * 1.296
d ≈ 8.2176 m
Therefore, the horizontal distance to the rocks at the moment the gull should let go of the clam is approximately 8.2176 m.
Next, let's calculate the speed with which the clam smashes into the rocks relative to the rocks. Since the clam is dropped from rest, the speed at which it hits the rocks can be calculated using the formula:
v = g * t
Where:
g = acceleration due to gravity (9.8 m/s^2)
t = time (1.296 s)
Plugging in the values, we get:
v = 9.8 * 1.296
v ≈ 12.709 m/s
Therefore, the speed of the clam smashing into the rocks relative to the rocks is approximately 12.709 m/s.
Finally, to find the speed of the clam relative to the gull when it smashes into the rocks, we can subtract the horizontal velocity of the gull from the speed of the clam hitting the rocks relative to the rocks.
Relative speed = Speed of the clam hitting the rocks relative to the rocks - Horizontal velocity of the gull.
Relative speed ≈ 12.709 m/s - 6.35 m/s
Relative speed ≈ 6.359 m/s
Therefore, the speed of the clam smashing into the rocks relative to the gull is approximately 6.359 m/s.