A sandal is dropped from the top of a 15.0-m-high mast on a ship moving at 1.75 m/s due south. Calculate the velocity of the sandal when it hits the deck of the ship: (a) relative to the ship and (b) relative to a stationary observer on shore. (c) Discuss how the answers give a consistent result for the position at which the sandal hits the deck.
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A sandal is dropped from the top of a 15.0-m-high mast on a ship moving at 1.75 m/s due south. Calculate the velocity of the sandal when it hits the deck of the ship: (a) relative to the ship and (b) relative to a stationary observer on shore. (c) Discuss how the answers give a consistent result for the position at which the sandal hits the deck.
To answer this question, we need to consider two components of the motion: the vertical motion of the sandal as it falls and the horizontal motion of the ship.
(a) To calculate the velocity of the sandal relative to the ship when it hits the deck, we only need to consider the vertical motion. We can use the equation of motion:
d = vi*t + (1/2)*a*t^2
Where:
d = displacement (vertical distance fallen) = 15.0 m
vi = initial vertical velocity = 0 (sandal is dropped)
t = time taken to fall
a = acceleration due to gravity = 9.8 m/s^2 (downward direction)
Solving for t, we have:
15.0 m = (1/2)*(9.8 m/s^2)*(t^2)
Simplifying the equation, we find:
t^2 = 15.0 m / (1/2 * 9.8 m/s^2) = 3.0612 s^2
Taking the square root of both sides, we get:
t ≈ 1.75 s
So, it takes approximately 1.75 seconds for the sandal to hit the deck. The velocity of the sandal relative to the ship is the vertical component of this motion, which is the speed the sandal is moving downward just before it hits the deck. This is given by:
vf = vi + a*t
= 0 + (9.8 m/s^2) * (1.75 s)
≈ 17.15 m/s (downward direction)
Therefore, the velocity of the sandal relative to the ship when it hits the deck is approximately 17.15 m/s downward.
(b) To calculate the velocity of the sandal relative to a stationary observer on the shore, we need to consider the horizontal motion of the ship. Since the ship is moving at a constant velocity of 1.75 m/s due south, the sandal will have the same velocity relative to the shore.
So, the velocity of the sandal relative to a stationary observer on the shore is 1.75 m/s due south.
(c) The consistent result for the position where the sandal hits the deck is because both the ship and the sandal have the same velocity when the sandal hits the deck. The horizontal motion of the ship does not affect the vertical motion of the sandal. Therefore, regardless of the ship's velocity, the sandal will fall straight down and hit the deck in the same position. The only difference is the sandal's horizontal position, which will be different based on the ship's movement.