A 9.4 kg weather rocket generates a thrust of 200 N. The rocket, pointing upward

Question

A 9.4 kg weather rocket generates a thrust of 200 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 520 N/m, is anchored to the ground.

(a) Initially, before the engine is ignited, the rocket sits at rest on top of the spring. How much is the spring compressed?

(b) After the engine is ignited, what is the rocket's speed when the spring has stretched 40 cm?

For comparison, what would be the rocket's speed after traveling this distance if it weren't attached to the spring?

Explanation / Answer

a)

mg = kx

x = (mg)/k

x = (9.4kg x 9.81m/s^2) / (520N/m)

x = 0.177 m

x = 17.7 cm

b)

PEi(spring) = 0.5(520)(0.177^2) = 8.1455

PEf(spring) = 0.5(520)(0.40^2) = 41.6

PEi(gravity) = 0

PEf(gravity) = 9.4(9.8)(0.177 + 0.40) = 53.15

W = 200(0.577) = 115.4

PE(spring) + KE + PE(gravity) = W

(41.6 - 8.1455) + 0.5(9.40)(v^2) + (53.15 - 0) = 115.4

33.4545 + 4.7v^2 + 53.15 = 115.4

4.7v^2 = 28.7955

v = 2.47 m/s

c)

if it's not attached to the spring then PEf(spring) = 0

PE(spring) + KE + PE(gravity) = W

(0 - 8.1455) + 0.5(9.40)v^2 + 53.15 = 115.4

4.7v^2 = 70.3955

v = 3.87 m/s

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