A paratrooper jumps freely from a plane for the first 100 m. When the parachute opens it will
provide an upward acceleration of 2.5 ms-2
. If he lands on the ground with a speed of 3 ms-1
,
a. how long is his time of flight.
b. What is the height of the plane.
To find the answers to these questions, we can use the equations of motion for uniformly accelerated motion.
Let's solve the problem step by step:
a. To find the time of flight, we need to find the time taken from the moment the parachute opens until the paratrooper lands on the ground. We can use the equation:
v = u + at
Where:
v = final velocity (3 m/s)
u = initial velocity (0 m/s, as the paratrooper is initially at rest)
a = acceleration (2.5 m/s^2, provided by the parachute)
t = time
Rearranging the equation to solve for time (t), we have:
t = (v - u) / a
Substituting the given values, we get:
t = (3 m/s - 0 m/s) / 2.5 m/s^2 = 1.2 s
Therefore, the paratrooper's time of flight is 1.2 seconds.
b. To find the height of the plane, we need to calculate the distance traveled by the paratrooper in the first 100 m before the parachute opens, and add that to the distance traveled while the parachute is open.
Let's calculate each part individually:
1. Distance covered before the parachute opens:
To find this distance, we can use the equation:
s = ut + (1/2)at^2
Where:
s = distance (100 m)
u = initial velocity (0 m/s)
t = time (unknown)
a = acceleration (9.8 m/s^2, assuming the paratrooper is close to the Earth's surface)
Rearranging the equation, we have:
100 m = 0.5 * 9.8 m/s^2 * t^2
Simplifying, we get:
t^2 = (100 m / (0.5 * 9.8 m/s^2))
t^2 = 20.41 s^2
Taking the square root of both sides, we find:
t = √20.41 s = 4.52 s
Therefore, the time taken to cover the first 100 m is approximately 4.52 seconds.
2. Distance covered while the parachute is open:
To find this distance, we can use the equation:
s = ut + (1/2)at^2
Where:
s = distance (unknown)
u = initial velocity (0 m/s, as the parachute opens and provides upward acceleration)
t = time (1.2 s)
a = acceleration (2.5 m/s^2, provided by the parachute)
Rearranging the equation, we have:
s = 0.5 * 2.5 m/s^2 * (1.2 s)^2
s = 1.8 m
Therefore, the distance covered while the parachute is open is 1.8 meters.
Finally, to find the height of the plane, we add the distances covered before and while the parachute is open:
Height = Distance before parachute opens + Distance while parachute is open
Height = 100 m + 1.8 m
Height = 101.8 m
Therefore, the height of the plane is approximately 101.8 meters.