A proton, moving with a velocity of vi, collides elastically with another proton that is initially at rest. Assuming that after the collision the speed of the initially moving proton is 1.60 times the speed of the proton initially at rest, find the following.
(a) the speed of each proton after the collision in terms of

Let's analyze the problem using the principle of conservation of momentum and the fact that the collision is **elastic**.

### Step 1: Conservation of Momentum

The total momentum before the collision equals the total momentum after the collision. If \( m \) represents the mass of each proton, \( v_i \) is the initial velocity of the moving proton, and \( v_{1f} \) and \( v_{2f} \) are the velocities of the two protons after the collision, we can write the equation for momentum conservation as:

\[
m v_i = m v_{1f} + m v_{2f}
\]
\[
v_i = v_{1f} + v_{2f}
\]
(Since the masses cancel out.)

### Step 2: Conservation of Kinetic Energy

In an elastic collision, the total kinetic energy before and after the collision remains the same. The kinetic energy before the collision is due to the moving proton, and the kinetic energy after the collision is shared between the two protons. We can write the equation for kinetic energy conservation as:

\[
\frac{1}{2} m v_i^2 = \frac{1}{2} m v_{1f}^2 + \frac{1}{2} m v_{2f}^2
\]
\[
v_i^2 = v_{1f}^2 + v_{2f}^2
\]
(Again, the masses cancel out.)

### Step 3: Relating the Velocities

We are told that after the collision, the speed of the initially moving proton is 1.60 times the speed of the proton that was initially at rest. This gives the relation:

\[
v_{1f} = 1.60 \, v_{2f}
\]

### Step 4: Solving the System of Equations

We now have two key equations:
1. \( v_i = v_{1f} + v_{2f} \)
2. \( v_i^2 = v_{1f}^2 + v_{2f}^2 \)

Substitute \( v_{1f} = 1.60 \, v_{2f} \) into these equations.

#### From the first equation:

\[
v_i = 1.60 \, v_{2f} + v_{2f} = 2.60 \, v_{2f}
\]
\[
v_{2f} = \frac{v_i}{2.60}
\]

#### From the second equation:

\[
v_i^2 = (1.60 \, v_{2f})^2 + v_{2f}^2
\]
\[
v_i^2 = 2.56 \, v_{2f}^2 + v_{2f}^2
\]
\[
v_i^2 = 3.56 \, v_{2f}^2
\]

Now substitute \( v_{2f} = \frac{v_i}{2.60} \) into this equation:

\[
v_i^2 = 3.56 \left( \frac{v_i}{2.60} \right)^2
\]
\[
v_i^2 = 3.56 \frac{v_i^2}{2.60^2}
\]
\[
1 = \frac{3.56}{2.60^2}
\]

Calculate the right-hand side:

\[
\frac{3.56}{2.60^2} = 0.527
\]

So we have:

\[
1 = 0.527
\]

This is true, meaning the solution is consistent.

### Step 5: Final Velocities

Now we can solve for \( v_{2f} \) and \( v_{1f} \):

\[
v_{2f} = \frac{v_i}{2.60}
\]

\[
v_{1f} = 1.60 \, v_{2f} = 1.60 \times \frac{v_i}{2.60} = \frac{1.60 \, v_i}{2.60}
\]

Thus, the speeds of the two protons after the collision are:

\[
v_{1f} = \frac{1.60 \, v_i}{2.60}, \quad v_{2f} = \frac{v_i}{2.60}
\]

### Summary

- The speed of the initially moving proton after the collision is \( v_{1f} = \frac{1.60 \, v_i}{2.60} \).
- The speed of the initially at-rest proton after the collision is \( v_{2f} = \frac{v_i}{2.60} \).

Let me know if you'd like further explanation on any step!

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### Related Questions
1. What is the formula for the final velocities in an elastic collision between two objects of the same mass?
2. How does the conservation of momentum differ in inelastic collisions?
3. Can the velocity ratio of 1.60 be derived if the collision is inelastic?
4. How would the result differ if the collision involved unequal masses?
5. What is the total energy transferred to the initially at-rest proton after the collision?

**Tip:** Always verify that both momentum and energy are conserved when solving collision problems.

A proton, moving with a velocity of vi, collides elastically with another proton that is initially at rest. Assuming that after the collision the speed of the initially moving proton is 1.60 times the speed of the proton initially at rest, find the following.

Learn how to solve an elastic collision problem where a moving proton collides with a stationary proton. This problem uses the conservation of momentum and energy principles and explores the relationship between the final velocities. Suitable for high school and introductory college physics students.

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