help please!! physics: proton question?

Question

A proton (mass 20.0 u) is shot at a speed of 10000000.0 m/s toward a gold target. The nucleus of a gold atom (mass 177.0 u) repels the proton and deflects it staight back toward the source with 98.0 percent of its initial speed. What is the recoil velocity of the gold nucleus?

Answer 1

you can use the equation of momentum which:
momentum before impact=momentum after impact
its (V1*m1+V2*m2)before impact=(V1*m1+V2*m2)after impact

Answer 2

This is just a conservation of momentum problem.
The initial momentum of the system must equal the final momentum of the system.

Momentum = mass * velocity
(for simplicity, we will just use atomic mass units (u) as the unit of mass to avoid the unnecessary complications of converting to kg)

We assume that the gold nucleus is initially at rest, so all of the initial momentum of the system comes from the 20.0 u positively charged particle being shot at the gold.
We are told the 20.0 u particle has a velocity of 1 E7 m/s.
From this information, we can calculate the initial momentum of the system as,
P_initial = 20.0 u * 1 E7 m/s
P_initial = 20 E7 u m/s.

After the collision occurs and the + particles rebounds off, its new velocity is 98% of its original speed in the opposite direction.
98% of 1 E7 m/s = 9.8 E6 m/s.
The + particles final momentum will be:
P_final_+ = 20.0 u * 9.8 E6 m/s
P_final_+ = -19.6 E7 u m/s.
The negative sign indicates that it is in the opposite direction as before.

We know the total final momentum must be equal to the initial momentum, and we also know that this must equal the sum of the final momentums of each particle. We just found the final momentum of the + particle…but this value alone does not equal the initial momentum. To this, we must add some value (the final momentum of the gold nucleus) to conserve momentum.
P_final = P_final_+ + P_final_gold = P_initial

Solving for the gold atom’s final momentum,
P_final_gold = P_initial – P_final_+
Plugging in,
P_final_gold = 20 E7 u m/s + 19.6 E7 u m/s
So we get a final momentum for the gold atom of,
P_final_gold = 39.6 u m/s, in the positive direction.

But we are asked to find the final velocity of the gold atom.
Do due this we go back to the original momentum formula,
P = mv
We know P, we know m, we just need to solve for v.
V = P / m
V = 39.6 u m/s / 177.0 u
V = .224 m/s in the positive direction.
The recoil velocity of the gold atom is extremely small compared to the initial and final speeds of the + particle… negligible in many cases.