# Maximum compression in a spring

#### Knfoster

1. Homework Statement

Two carts of equal mass, m = 0.300 kg, are placed on a frictionless track that has a light spring of force constant k = 48.9 N/m attached to one end of it. The red cart is given an initial velocity of v0 = 3.13 m/s to the right, and the blue cart is initially at rest. If the carts collide elastically, find the magnitude of the velocity of the red cart just after the first collision. Find the magnitude of the velocity of the blue cart just after the first collision.
Find the maximum compression in the spring.

2. Homework Equations
pe=1/2*k*change in x
m1v1=m2v2

3. The Attempt at a Solution
THe velocity of the red cart after the first collision=0 m/s
the velocity of the blue cart after the first collison=3.13 m/s
I don't know how to find the maximum compression in the spring... Please help. Thanks.

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#### J Hill

Okay, I have done some work with this before, but if I'm wrong, I hope someone else can correct me. When two objects collide elastically, they naturally compress and kind of stick together momentarily. In this case, the collision acts like an inelastic collision, which causes some of the energy of the system to be stored in the spring. So, if you treat that system during the collision as inelastic you get the velocity of the two carts (together) to be 1.57 m/s and the KE to be 0.735 J. The total energy of the system is 1.47 J. Using these values with the conservation of energy, you should be able to calculate the maximum compression in the spring.

#### Dr.D

The maximum compression of the spring should occur at the point when the two bodies come to a common velocity. At this point, one body will have lost velocity and the other will have gained, and they will be at their point of closest approach to each other, ie, the maximum compression of the spring. Thereafter, the spring begins to expand and the velocities begin to separate.

#### srabate

If I'm interpreting the problem correctly, I believe you just have to find the compression of the spring when all of the kinetic energy of the blue cart is put into the compression of the spring.

As you said.. after the collision:
The velocity of the red cart is 0 m/s
The velocity of the blue cart is 3.13 m/s

Thus you know the total energy of the system before the blue cart hits the spring. This is equal to the energy
of the system after it hits the spring.
$$\frac{1}{2}mv^{2}=\frac{1}{2}kx_{max}^{2}$$

(the velocity of the blue cart is 0 at the max compression... that is why there is no $$\frac{1}{2}mv^{2}$$ term on the RHS)

Hope it helps!

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