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?
b) 4.51 m/s
To find the horizontal distance to the rocks when the gull should let go of the clam, we can use the equation:
d = v*t
Where:
d is the horizontal distance
v is the horizontal velocity
t is the time it takes for the clam to fall
In this case, the horizontal velocity of the gull is 6.35 m/s and the time it takes for the clam to fall can be found using the equation:
t = sqrt(2h/g)
Where:
h is the height from which the clam is dropped (8.17 m)
g is the acceleration due to gravity (9.8 m/s^2)
Plugging in the values, we have:
t = sqrt(2*8.17/9.8) = 1.02 s
Substituting this value into the first equation, we can find the horizontal distance:
d = 6.35 * 1.02 = 6.47 m
Therefore, the gull should let go of the clam when it is approximately 6.47 meters above the rocks.
To find the speed at which the clam smashes into the rocks relative to the rocks, we can use the equation:
v_rock = sqrt(v_gull^2 + v_clam^2)
Where:
v_rock is the speed of the clam relative to the rocks
v_gull is the speed of the gull (6.35 m/s)
v_clam is the speed of the clam relative to the gull
Since the clam is dropped from the gull at rest relative to the gull, its speed relative to the gull is 0. Therefore, we plug in the values:
v_rock = sqrt(6.35^2 + 0^2) = 6.35 m/s
Therefore, the clam smashes into the rocks with a speed of 6.35 m/s relative to the rocks.
To find the speed at which the clam smashes into the rocks relative to the gull, we can subtract the speed of the gull from the speed of the clam relative to the rocks:
v_clam = v_rock - v_gull = 6.35 - 6.35 = 0 m/s
Therefore, the clam smashes into the rocks with a speed of 0 m/s relative to the gull.
To determine the horizontal distance to the rocks, we need to first find the time it takes for the clam to hit the ground. This can be done by using the equation:
y = v0y * t + (1/2) * a * t^2
Where:
- y is the vertical displacement (equal to the height above the ground)
- v0y is the initial vertical velocity (equal to 0 since the gull is flying horizontally)
- a is the acceleration due to gravity (approximately equal to 9.8 m/s^2)
- t is the time it takes for the clam to hit the ground
Plugging in the values, we have:
8.17 m = 0 * t + (1/2) * 9.8 m/s^2 * t^2
Simplifying the equation, we get:
4.9 t^2 = 8.17
Solving for t, we find:
t = sqrt(8.17 / 4.9)
t ≈ 1.47 seconds
Now that we have the time it takes for the clam to hit the ground, we can find the horizontal distance by multiplying the horizontal velocity of the gull (6.35 m/s) by the time:
Horizontal distance = 6.35 m/s * 1.47 s
Horizontal distance ≈ 9.34 meters
So, the gull should let go of the clam when it is approximately 9.34 meters away from the rocks.
To determine the speed at which the clam smashes into the rocks, we need to find the vertical velocity of the clam just before it hits the ground. We can use the equation:
v = v0 + a * t
Where:
- v is the final velocity (which is equal to the speed at which the clam smashes into the rocks)
- v0 is the initial vertical velocity (equal to 0 since the gull is flying horizontally)
- a is the acceleration due to gravity (approximately equal to 9.8 m/s^2)
- t is the time it takes for the clam to hit the ground (which we found to be approximately 1.47 seconds)
Plugging in the values, we have:
v = 0 + 9.8 m/s^2 * 1.47 s
v ≈ 14.42 m/s
So, the clam smashes into the rocks with a speed of approximately 14.42 m/s relative to the rocks.
To determine the speed at which the clam smashes into the rocks relative to the gull, we need to consider the horizontal velocity of the gull (6.35 m/s). Since the gull is flying horizontally, the horizontal velocity remains unchanged. Therefore, the speed at which the clam smashes into the rocks relative to the gull is also 6.35 m/s.