To answer these questions, we need to consider the motion of the man and the helicopter. Let's break it down step-by-step:
a. To determine the speed of the man just before he lands on the pillow, we need to find the velocity at that moment. Since the helicopter is traveling upward with a velocity of 12 m/s, its velocity will contribute to the man's velocity when he lets go. The man's velocity just before he lands can be calculated by subtracting the velocity of the helicopter from his own initial velocity.
To find the man's initial velocity, we can use the equation of motion:
v = u + at
Where:
v = final velocity (unknown)
u = initial velocity (given as 12 m/s)
a = acceleration (acceleration due to gravity, approximately -9.8 m/s^2, considering downward direction)
t = time (unknown)
Since the motion is vertical, the acceleration due to gravity acts in the opposite direction (downward), so it is taken as negative. We can assume the positive direction is upward, so we'll use a negative sign for acceleration.
When the man lets go of the rope, he experiences free fall. We can use the equation for displacement:
s = ut + (1/2)at^2
To find s (displacement), we consider the final height above the pillow (s = -20 m) and solve for t.
Since the man and the helicopter are at the same height when he lets go, we can set s = 0. Now the equation becomes:
0 = 12t + (1/2)(-9.8)t^2
Simplifying the equation:
4.9t^2 - 12t = 0
Factoring out t:
t(4.9t - 12) = 0
This gives us two possibilities:
t = 0 (not relevant in this case)
t = 12/4.9 ≈ 2.45 seconds
Therefore, the man is in the air for approximately 2.45 seconds after he lets go of the rope.
To find the speed of the man just before he lands, we substitute the value of t into the equation "v = u + at":
v = 12 + (-9.8)(2.45)
v ≈ 12 - 24.01
v ≈ -12.01 m/s (negative sign indicates downward direction)
b. We have already calculated that the man is in the air for 2.45 seconds. This is the time from when he lets go of the rope until he lands on the pillow.
c. To find the greatest height reached by the man above the ground, we need to calculate his maximum displacement (or maximum height). We can use the equation for displacement again:
s = ut + (1/2)at^2
Since we know u (initial velocity) and t (time), we can substitute these values into the equation:
s = 12(2.45) + (1/2)(-9.8)(2.45)^2
Simplifying:
s = 29.4 - 30.1475
s ≈ -0.7475 m
Since the displacement is negative, it means the highest point the man reached is below his initial position. In this case, the pillow is situated 20 m below the man's initial position (as stated in the question). Therefore, the man's greatest height is above the ground and can be calculated by adding his displacement to the height of the pillow. So,
Greatest height reached = (-0.7475) + 20 ≈ 19.25 m
Therefore, the greatest height reached by the man above the ground is approximately 19.25 m.
d. To find the distance between the helicopter and the man three seconds after he lets go of the rope, we need to calculate the horizontal distance traveled by the helicopter in three seconds. Since the helicopter is moving directly upward and the horizontal velocity remains constant, the distance traveled will be equal to the horizontal velocity multiplied by the time taken.
The horizontal distance is given by:
Distance = (horizontal velocity) * time
= (0 m/s) * 3 s
= 0 m
Therefore, the distance between the helicopter and the man three seconds after he lets go of the rope is zero meters. This means the helicopter and the man are at the same horizontal position.