A block of massslides down a 30.0° incline which is 3.60 m high. At the bottom, it strikes a block of masswhich is at rest on a horizontal surface. (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine (a) the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go.

This is a classical physics problem involving energy conservation and an elastic collision. Here's a step-by-step breakdown of how to approach it:

### Part (a): Determine the speeds of the two blocks after the collision.

Let's denote:
- \( m_1 \) as the mass of the first block (the one sliding down the incline),
- \( m_2 \) as the mass of the second block (the one initially at rest),
- \( v_1 \) as the velocity of block \( m_1 \) just before the collision,
- \( v_2 \) as the velocity of block \( m_2 \) just before the collision (which is \( 0 \) since it's at rest),
- \( v_1' \) as the velocity of block \( m_1 \) after the collision,
- \( v_2' \) as the velocity of block \( m_2 \) after the collision.

#### Step 1: Find the speed of \( m_1 \) just before the collision
The block \( m_1 \) slides down the incline without friction, so we can use **conservation of mechanical energy**.

The potential energy at the top is entirely converted into kinetic energy at the bottom:

\[
m_1 g h = \frac{1}{2} m_1 v_1^2
\]

Solve for \( v_1 \):

\[
v_1 = \sqrt{2gh}
\]

Given:
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( h = 3.60 \, \text{m} \) (height of the incline),

\[
v_1 = \sqrt{2 \times 9.8 \times 3.60} = \sqrt{70.56} \approx 8.40 \, \text{m/s}
\]

So, the speed of \( m_1 \) just before the collision is approximately \( 8.40 \, \text{m/s} \).

#### Step 2: Apply the equations for an elastic collision
For a **perfectly elastic collision**, two key principles are used:
1. **Conservation of momentum**:
\[
m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'
\]
Since \( v_2 = 0 \) (block 2 is initially at rest), the equation simplifies to:
\[
m_1 v_1 = m_1 v_1' + m_2 v_2'
\]

2. **Conservation of kinetic energy**:
\[
\frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_1 {v_1'}^2 + \frac{1}{2} m_2 {v_2'}^2
\]

#### Solving for the final velocities:
In an elastic collision, there's a known formula for the final velocities of two colliding objects:

\[
v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1
\]

\[
v_2' = \frac{2 m_1}{m_1 + m_2} v_1
\]

We can substitute the known values of \( v_1 \) once we know the masses \( m_1 \) and \( m_2 \).

### Part (b): How far back up the incline the smaller mass will go

After the collision, block \( m_1 \) will travel back up the incline. We can use conservation of energy again. The kinetic energy of block \( m_1 \) after the collision will convert back into potential energy as it moves up the incline.

\[
\frac{1}{2} m_1 {v_1'}^2 = m_1 g h_{\text{max}}
\]

Solve for \( h_{\text{max}} \):

\[
h_{\text{max}} = \frac{{v_1'}^2}{2g}
\]

Once \( v_1' \) is calculated from part (a), we can find how high the block rises.

### Final Steps:
- To solve for the final velocities and height, we'd need the values of \( m_1 \) and \( m_2 \).
- If these values are given, we can plug them into the equations above for specific results.

---

Would you like to provide values for the masses, or do you need further details on the general case?

A block of mass slides down a 30.0° incline which is 3.60 m high. At the bottom, it strikes a block of mass which is at rest on a horizontal surface. (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine (a) the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go.

Learn how to solve a physics problem involving an elastic collision between two blocks, one sliding down an incline. Using conservation of energy and momentum, this problem covers how to calculate the velocities of the blocks post-collision and how far the smaller block travels back up the incline. Ideal for undergraduate physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation