A spring has a relaxed length of 34 cm (0.34 m) and its spring stiffness is 12 N/m. You glue a 72 gram block (0.072 kg) to the top of the spring, and push the block down, compressing the spring so its total length is 12 cm. You make sure the block is at rest, then at time t = 0 you quickly move your hand away. The block begins to move upward, because the upward force on the block by the spring is greater than the downward force on the block by the Earth. Calculate approximately y vs. time for the block during a 0.21-second interval after you release the block, by applying the Momentum Principle in three steps each of 0.07-second duration.

To avoid buildup of small errors causing you to lose credit, in Step 2 we use your answers to Step 1 even if they are not correct, and in Step 3 we use your answers to Step 2 even if they are not correct.

We will only consider the y components in the following calculations, because there is no change in x or z.

STEP 1

Force: Just after releasing the block, calculate the force exerted on the block by the spring, the force exerted on the block by the Earth, and the net force:

F_spring,y =  N

F_Earth,y =  N

F_net,y =  N

Momentum update: Just after releasing the block, the momentum of the block is zero. Approximate the average net force during the next time interval by the force you just calculated. At t = 0.07 seconds, what will the new momentum and velocity of the block be?

p_y =  kg · m/s

v_y =  m/s

Position update: Initially the bottom of the block is at y = 0.12 m. Approximating the average velocity in the first time interval by the final velocity, what will be the new position of the bottom of the block at time t = 0.07 seconds?

y =  m

STEP 2

Force: At the new position, calculate the force exerted on the block by the spring, the force exerted on the block by the Earth, and the net force:

F_spring,y =  N

F_Earth,y =  N

F_net,y =  N

Momentum update: Approximate the average net force during the next time interval by the force you just calculated. At time t = 2 × 0.07 = 0.14 seconds, what will the new momentum and velocity of the block be?

p_y =  kg · m/s

v_y =  m/s

Position update: Approximating the average velocity in the second time interval by the final velocity, what will be the new position of the bottom of the block at time t = 2 × 0.07 = 0.14 seconds?

y =  m

STEP 3

Force: At the new position, calculate the force exerted on the block by the spring, the force exerted on the block by the Earth, and the net force:

F_spring,y =  N

F_Earth,y =  N

F_net,y =  N

Momentum update: Approximate the average net force during the next time interval by the force you just calculated. At time t = 3 × 0.07 = 0.21 seconds, what will the new momentum and velocity of the block be?

p_y =  kg · m/s

v_y =  m/s

Position update: Approximating the average velocity in the third time interval by the final velocity, what will be the new position of the bottom of the block at time t = 3 × 0.07 = 0.21 seconds?

y =  m