A mass M = 5 kg moves with an initial speed v = 3 on a level frictionless air track. The mass is initially a distance D = 0.250 m away from a spring with k = 844 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.
I have this part..just need help with part b

(b) Find the total elapsed time until the mass returns to its starting point. (Hint: The mass undergoes a partial cycle of simple harmonic motion while in contact with the spring.)

figure the period of the spring (see the spring harmonic oscillator equations). Then, the time on the spring is one quarter cycle (compressing it), then the spring accelerates the mass to some velocity and as the spring slows, the mass leaves. Max velocity occurs in the spring at the unstretched point, so in my mind, 1/2 cycle total.

The law of conservation of energy

KE (of the mass) = PE (of the spring)
mv^2/2 = kx^2/2
d=x = sqrt (mv^2/k)=sqrt(5•9/844)=0.23 m

T = 2•π•sqrt(m/k) =2•π•sqrt(5/844) = 0.484 s
The time of the mass motion with spring is T/2 =0.242 s.
The time of uniform motion before hitting the spring and after leaving thespring is
t =D/v =0.25/3=0.0833 s.

Therefore, the total time is 0.242 + 2•0.0833=0.409 s

To find the total elapsed time until the mass returns to its starting point, we need to consider the motion of the mass during two different phases: the phase where it is compressed against the spring and the phase where it is moving freely without any contact with the spring.

Let's break down the problem into steps:

Step 1: Calculate the maximum velocity of the mass when it is compressed against the spring using the conservation of mechanical energy.

From the conservation of mechanical energy, we know that the initial kinetic energy of the mass will be equal to the potential energy stored in the compressed spring.

Initial kinetic energy = 1/2 * M * v^2
Potential energy in the compressed spring = 1/2 * k * d^2

Since the mass reverses direction and returns to its starting point, the final kinetic energy is also 1/2 * M * v^2.

Therefore, we can equate the initial and final kinetic energies to the potential energy stored in the compressed spring:

1/2 * M * v^2 = 1/2 * k * d^2

We can rearrange this equation to solve for the maximum distance the spring is compressed, d:

d = sqrt((M * v^2) / k)

Once we have found the value of d, we can move on to the next step.

Step 2: Determine the time taken to reach the maximum compression.

Since the mass undergoes simple harmonic motion while in contact with the spring, we can use the equation for the period of a mass-spring system:

T = 2π * sqrt(M / k)

The time taken to reach the maximum compression is half the period, so:

t1 = (1/2) * T = π * sqrt(M / k)

Step 3: Calculate the distance traveled by the mass while moving freely without any contact with the spring.

The mass is initially a distance D away from the spring. It compresses the spring a maximum distance d, and then in the opposite direction with the same speed v. So, the total distance traveled without contact with the spring is:

2(D + d)

Step 4: Determine the time taken to cover the distance without contact with the spring.

The time taken to cover a distance at constant speed is given by:

t2 = (distance) / (speed) = 2(D + d) / v

Step 5: Find the total elapsed time.

The total elapsed time until the mass returns to its starting point is the sum of the times taken in steps 2 and 4:

Total elapsed time = t1 + t2 = π * sqrt(M / k) + 2(D + d) / v.

Now you have the formula to find the total elapsed time until the mass returns to its starting point. Plug in the given values of M, v, D, and k to calculate the answer.