A tallest vocano in the Solar System is the 24 km tall Martian volcano, Olympus Mons. Assume an astronaut drops a ball off the rim of the crater and that the free fall acceleration remains constant throughout the ball's 24 km fall at a value of 3.7m/s^2. Find (a) the time for the ball to reach the crater floor and (b) the velocity with which it hits.
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To find the time it takes for the ball to reach the crater floor, we can use the equations of motion. The equation we'll be using for this case is:
h = (1/2) * g * t^2
where h is the height, g is the acceleration due to gravity, and t is the time it takes for the ball to reach the height h.
In this case, the height h is 24 km, which is equivalent to 24,000 meters. The acceleration due to gravity, g, remains constant at 3.7 m/s^2.
(a) Finding the time for the ball to reach the crater floor:
We'll rearrange the equation as follows:
t^2 = (2 * h) / g
Substituting the given values:
t^2 = (2 * 24,000) / 3.7
t^2 = 12,973.97
To find t, we take the square root of both sides:
t ≈ √12,973.97
t ≈ 113.97 seconds
Therefore, it takes approximately 113.97 seconds for the ball to reach the crater floor.
(b) Finding the velocity with which it hits:
We can use another equation of motion to calculate the velocity:
v = g * t
Substituting the values:
v = 3.7 * 113.97
v ≈ 419.89 m/s
Therefore, the velocity at which the ball hits the crater floor is approximately 419.89 m/s.