A sailor climbs to the top of the mast, 12.55 m above the deck, to look for land while his ship moves steadily forward through calm waters at 4.63 m/s. Unfortunately, he drops his spyglass to the deck below.
Where does it land with respect to the base of the mast below him?
The time (T) that it takes the spyglass to go in the x-direction is the same amount of time that it takes to travel in the y-direction. Since in the Y-direction the initial velocity is 0 and acceleration is equal to gravity (9.8m/s^2), use the equation D=ViT+1/2(aT^2). Once you plug in the values, you should get a simplified equation of D=1/2(9.8m/s^2)(T)^2. rearranging it to solve for T should result in the equation looking as followed: T=[2D/(9.8m/s^2)]^1/2. Since velocity is constant in the x-direction, use D=VT and solve for the distance in the x-direction using the value for T that you calculated. D=(4.63m/s)*T.
To determine where the spyglass lands with respect to the base of the mast, we need to consider the horizontal motion of the ship and the vertical motion of the spyglass.
First, let's calculate the time it would take for the spyglass to fall to the deck. We can use the formula for free-fall motion:
h = (1/2) * g * t^2
Where:
h is the height (12.55 m)
g is the acceleration due to gravity (9.8 m/s^2)
t is the time
Rearranging the formula to solve for time:
t = sqrt(2 * h / g)
Substituting the given values into the formula:
t = sqrt(2 * 12.55 / 9.8) = sqrt(2.551 = 1.598) seconds (rounded to three decimal places)
Since the sailor drops the spyglass from the top of the mast, it has no initial horizontal velocity. Meanwhile, the ship moves forward with a velocity of 4.63 m/s.
The horizontal distance the spyglass would travel in that time can be calculated using the formula:
d = v * t
Where:
d is the horizontal distance
v is the horizontal velocity (4.63 m/s)
t is the time (1.598 s)
Substituting the given values into the formula:
d = 4.63 * 1.598 = 7.393 meters (rounded to three decimal places)
Therefore, the spyglass will land 7.393 meters horizontally away from the base of the mast.