To determine the required horizontal velocity of the rescue line, we can consider the horizontal motion as it is thrown horizontally from the bridge of the ship. In this case, we can neglect the effects of air resistance.
The horizontal distance traveled by the rescue line is given as 30 meters. We can use the equation for horizontal motion:
d = v_x * t
where:
- d is the horizontal distance (30 meters),
- v_x is the horizontal velocity (the value we want to find),
- t is the time of flight.
Since the rescue line is thrown horizontally, the initial vertical velocity is zero. Therefore, we can use the vertical motion equation to find the time of flight:
h = (1/2) * g * t²
where:
- h is the vertical displacement (the height difference between the bridge and the lifeboat, which is 10 meters),
- g is the acceleration due to gravity (-9.8 m/s²),
- t is the time of flight.
Substituting the values into the equation:
10 = (1/2) * (-9.8) * t²
Simplifying the equation:
20 = -4.9 * t²
Dividing both sides by -4.9:
t² = -20 / -4.9
t² ≈ 4.08
Taking the square root of both sides:
t ≈ √4.08
t ≈ 2.02 seconds
Now that we have the time of flight, we can find the horizontal velocity by rearranging the horizontal motion equation:
v_x = d / t
v_x = 30 / 2.02
v_x ≈ 14.85 m/s
Therefore, the rescue line should be thrown with a horizontal velocity of approximately 14.85 m/s.