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?
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 can start by analyzing the motion of the gull horizontally. Since the gull is flying horizontally, its horizontal velocity remains constant at 6.35 m/s. We can use the equation for horizontal distance (d) to find the time it takes for the clam to hit the ground:
d = v * t
Where:
d = horizontal distance
v = horizontal velocity of the gull
t = time taken
Rearranging the equation, we get:
t = d / v
Since the vertical distance between the gull and the ground is given as 8.17 m, we can use this information to find the time taken for the clam to hit the ground vertically. We can use the equation for vertical distance (h) and time (t) in free fall with gravity (9.8 m/s²):
h = (1/2) * g * t²
Where:
h = vertical distance
g = acceleration due to gravity
t = time taken
Rearranging the equation, we get:
t = sqrt(2 * h / g)
Substituting the given values, we find:
t = sqrt(2 * 8.17 m / 9.8 m/s²) ≈ 1.421 s
Now, we can substitute the value of time (t) into the equation for horizontal distance (d) to find the horizontal distance to the rocks:
d = v * t
d = 6.35 m/s * 1.421 s
d ≈ 9.031 m
So, the horizontal distance to the rocks when the gull should let go of the clam is approximately 9.031 meters.
To find the speed at which the clam smashes into the rocks relative to the rocks, we can consider the vertical motion of the clam. Since the clam is being dropped from rest, it will fall freely under the influence of gravity. The speed at which it hits the rocks can be found using the equation for velocity (v) in free fall:
v = g * t
Substituting the value of time (t) into the equation, we get:
v = 9.8 m/s² * 1.421 s
v ≈ 13.919 m/s
So, the clam smashes into the rocks with a speed of approximately 13.919 m/s relative to the rocks.
To find the speed at which the clam smashes into the rocks relative to the gull, we can consider their horizontal motion. Since the gull and the clam are moving horizontally at the same velocity, the speed of the clam relative to the gull will be the same as the horizontal velocity of the gull:
v = 6.35 m/s
So, the clam smashes into the rocks with a speed of 6.35 m/s relative to the gull.