Any help would be greatly appreciated?

Question

Box 1 of mass m1=5.0kg starts from rest in point A and slides down a frictionless ramp of height h=0.50m. In point B, the surface becomes horizontal and rough, with a coefficient of kinetic friction u=0.1. In point B, Box 1 collides head-on elastically with the stationary Box 2 of mass m2=7.0kg. After collision, the Box 1 slides back up on the ramp, and Box 2 travels a horizontal distance d=0.50m. In point C, Box 2 hits a long spring with elastic constant k=350 N/m. The spring is compressed by the incoming box (the surface under the spring is frictionless).

a) Calculate the speed, VB, of Box 1 at the base of the ramp, just before the collision.
b) Calculate the velocities of the two boxes immediately after the head-on elastic collision.
c) How high up the incline does Box 1 climb after the collision?
d) What is the kinetic energy of Box 2 immediately before striking the spring?
e) Calculate the maximum compression of the spring.

Answer 1

Talk about a kitchen sink problem, this one has it all. The first part is all about conservation of energy:

The formula for gravitational energy is:
Eg = M x g x H

where H is the vertical distance and g is the acceleration due to gravity (9.8 m/s on the surface of the Earth)

The formula for kinetic energy is:

Ek = (1/2) x M x V^2

where V is the velocity. (This assumes only linear velocity - things get more complicated when there is rolling, etc.)

When the block starts from rest at the top of the slide, it has a certain potential energy.

As it slides down, this potential energy is converted to kinetic energy. Since the slide is frictionless, the conversion is lossless.

When the block has slid down 0.5 meters of height, it has converted the potential energy associated with that height to kinetic energy. So:

M x g x 0.5 = (1/2) x M x V^2

and you can compute the VB which is the answer to part a.

If the collision is at the bottom of the ramp, then you already know the total energy of the collision. Since the collision is elastic, no energy is lost. Since there is no change of height, the only energy in question is kinetic.

Ei = Eo
Eo = Eo1 + Eo2
Eo1 = (1/2) x M1 x V1^2
Eo2 = (1/2) x M2 x V2^2

where M1 is the mass of box 1, M2 is the mass of box 2, V1 is the resulting velocity for box 1, etc.

But a collision also conserves momentum. Linear momentum is just mass times velocity:

Pi = Po
Pi = M1 x VB
Po = M1 x V1 + M2 x V2

So you have two unknowns, V1 and V2 and two equations, one from conservation of energy (a quadratic) and one from conservation of momentum.

That takes care of part b.

Part c is still conservation of energy. Box 1 starts at the bottom of the ramp with known velocity and hence known kinetic energy. When it stops, all its kinetic energy has been converted to potential energy so:

M1 x g x H1 = (1/2) x M1 x V1^2

which is a linear equation in the desired answer H1.

When box 2 moves along a surface with friction, there is a drag D which slows it down

D = U x N

where:

U is the coefficient of kinetic friction
N is the normal force

In this case, with only gravity operating, N is just the force of gravity or the weight of box 2.

So drag D generates an acceleration (the slowing down) of box 2. The general equation for travel under constant acceleration is:

X = Vi x T + (1/2) x A x T^2

Here we know the distance traveled X = 0.5 meters, the initial velocity Vi (just V2 from above), and the acceleration A due to drag, so we can compute T.

Given T, Vi, and A, we can compute the velocity at point C:

Vc = Vi - A x T

Given Vc and M2, we can compute the kinetic energy of block 2 at point C to get the answer to part d.

Now comes another conservation of energy section.

The formula for the energy in a simple spring is

Es = (1/2) x K x D^2

where:

K is the spring constant of the spring (assumed constant for simple springs)

D is the displacement of the spring (for a linear spring the change in the length of the spring)

When the spring is fully compressed, it is because the velocity has gone to 0 so all the kinetic energy has become spring energy.

Given the total energy, the mass M2, and the spring constant, you can compute the displacement of the spring, for the answer to part e.

Answer 2

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