Question

Colby participates in a rocket-launch competition in her high school. The goal is to launch a rocket with mass m = 2.3 kg [5 lbs] to a height no less than 50-m (164 ft) above ground.

a) What is the minimum momentum of the rocket as it leaves the launch pad?

b) Colby figured out three possible propulsion mechanisms that differ by the generated force F (“thrust”) and the time interval ∆t during which the force acts on the rocket [“burn time”]: Force F [N] Time interval ∆t [s] Propulsion #1 161 0.41 Propulsion #2 233 0.37 Propulsion #3 532 0.12 What is Colby’s best choice for the propulsion mechanism? Explain your reasoning!

c) Colby is excited about the successful launch of her rocket but is disappointed because the rocket is in air for only t1 = 3.1 s after the launch until it reaches the highest point. Explain Colby’s disappointment and calculate the air resistance acting on the rocket! Assume that the air resistance is constant.

d) Does Colby’s rocket meet the requirement of the contest?

e) The rocket returns to the ground at the time t2 = 6.5 s measured from the time of the launch. Calculate the impulse imparted on the rocket during the flight form the highest back to the ground. Find the velocity vector!] of the rocket just before it lands on the ground!

Answer 1

(a) To travel a vertical distance 50 m , required projection speed u of rocket is obtained from the formula

where v is final speed which is zero when rocket reaches maximum height , g is acceleration due to gravity and h is vertical distance

Hence we get u from above formula,

required momentum = mass velocity = 2.3 31.3 = 72 kg-m/s

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(b) Impulse given by three propulsion mechanism is calculated as given below

Impulse = force time

#1 :-   161 N 0.41 s = 66.01 kg-m/s

#2 :-    233 N 0.37 s = 86.21 kg-m/s

#3 :-   532 N 0.12 s = 63.84 kg-m/s

propulsion scheme #2 gives the momentum that satisfies the minimum requirement

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(c) Let us assume propulsion scheme #2 is used to launch the rocket.

initial speed u = momentum / mass = 86.21/ 2.3 = 37.5 m/s

Expected time in air from launch time to maximum height = u/g = 37.5/9.8 = 3.8 s

But the rocket was in air only for a time 3.1 s

The difference is because of air resistance . Air resistance gives retardation in addition to gravity .

Hence rocket has travelled less time.

Air resistance is calculated by finding the true retardation as given below

retardation = Initial speed / time = 37.5/3.1 = 12.1 m/s2

Retardation due to only air resistance = ( 12.1 - 9.8 ) m/s2 = 2.3 m/s2

Air resistance = mass retardation = 2.3 2.3 = 5.29 N

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(d) Maximum height h reached by rocket ,

Since the requirement of maximum height is 50 m , Colby's rocket meets the requirement of contest

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(e) Total time of flight = 6.5 s

Time to reach maximum height = 3.1 s

Hence travelling time of downward motion = ( 6.5 - 3.1 ) = 3.4 s

If maximum height is 58.11 m and time taken to reach ground is 3.4 s , then acceleration a of the motion is calculated as given below

impulse given to rocket during downward motion = Force time = mass acceleration time

impulse = 2.3 10.05 3.4 = 78.6 kg m/s

velocity v when the rocket reaching ground ,

v = acceleration time = 10.05 3.4 = 34.17 m/s

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