A mass M = 0.491 kg moves with an initial speed v = 3.41 m/s on a level friction

Question

A mass M = 0.491 kg moves with an initial speed v = 3.41 m/s on a level frictionless air track. The mass is initially a distance D = 0.213 m away from a spring with k = 849 N/m, which is mounted rigidly at one end of the air track. The mass compresses the spring a maximum distance d, before reversing direction. After bouncing off the spring, the mass travels with the same speed v, but in the opposite direction.

a) Determine the maximum distance that the spring is compressed.


b) Find the total elapsed time until the mass returns to its starting point.

Explanation / Answer

a) 1/2*Mv2 = 1/2*kx2

So, 0.491*3.412 = 849*x2

x = 0.082 m

b) During spring compression and expansion, average force = kx/2

avergae deceleration and acceleration, a = (kx/2)/M = (849*0.082/2)/0.491 = 70.9 m/s2

Time during compression (or expansion) = v/a = 3.41/70.9 = 0.048 s

Total time during compression and expansion = 2*0.048 = 0.096 s

Time needed to reach and return from initial position is 2*D/v = 2*0.213/3.41 = 0.125 s

Total time = 0.125+0.096 = 0.221 s

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