A rocket blasts off vertically from rest on the launch pad with an upward accele

Question

A rocket blasts off vertically from rest on the launch pad with an upward acceleration of 2.10m/s2 . At 30.0safter blastoff, the engines suddenly fail, which means that the force they produce instantly stops.

a)How high above the launch pad will the rocket eventually go?

b)Find the rocket's velocity at its highest point.

c)Find the magnitude of the rocket's acceleration at its highest point.

d)Find the direction of the rocket's acceleration at its highest point. (horizontal, downward or upward)

e)How long after it was launched will the rocket fall back to the launch pad?

f)How fast will it be moving when it does so?

Explanation / Answer

1. To find out how far the rocket will go, first find how high the rocket goes with help from the engine, then find out how high the rocket goes after the engine is cut off.
So:
when engine is on:
a= 2.1m/s^2
vi = 0m/s
t = 30s
d = vit + 1/2at^2
d = 0 + 1/2x2.1x30^2
d= 945m

find final velocity, so that you can answer the next part (final velocity of rocket while engine is on becomes initial velocity when engine is off)
vf^2 = vi^2 + 2ad
vf^2 = 0 + 2x2.1x945
vf= 63m/s

when engine is off, but rocket is still going up:
a = -9.81m/s^2 (force of gravity)
vi = 63m/s (vf from before becomes vi)
vf = 0m/s (when the rocket reaches max height, the final velocity will be 0)
to find d, use formula vf^2= vi^2 + 2ad
0 = 63^2 + 2x(-9.81)d
solve the equation to get d= 202.3m
so max distance would be 945 + 202.3 = 1147.3m

2. Magnitude of rocket's acceleration at its highest point is going to be -9.81m/s^2, or the acceleration due to the force of gravity- remember, that the engines are off, so there isn't any other force acting on the rocket save for the force of gravity.

3. You already know the time duration while the rocket's engines were working, but now we need to find the time it takes while the rocket is still moving upwards with the engines off, and find how long does it take for the rocket to get back down.
To find how long it takes for the rocket to move upwards while engine is off:
d= 202.3m
vi=63m/s
a= -9.81m/s^2
vf=0 (when the rocket reaches the top, vf is 0)
vf= vi + at
0=63 - 9.81t
Solve to get t= 6.42s
To find the time it takes for the rocket to get from the top to the bottom:
d= -1147.3m (top to bottom)
vi=0m/s
a=-9.81m/s^2
d= vit + 1/2at^2
-1147.3= 0+ 1/2x(-9.81)t^2
solve for t to get t=15.3s
so add up all the times to get the total time the rocket was in the air: 30 + 6.42 + 15.3 = 51.7s

4. use the formula vf=vi + at to find the final velocity
where vi = 0
a = -9.81m/s^2
t= 15.3s
vf = 0 + (-9.81)x15.3
vf = -150.1m/s

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