Conservationof Momentum Worksheet | PDF - Free Printable

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Here’s a breakdown of what appears to be in the worksheet and how to solve such problems:

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Problem 1: Elastic Collision (Two objects moving toward each other)

Given:
- Object 1: mass = 1 kg, velocity = +2 m/s (to the right)
- Object 2: mass = 2 kg, velocity = -1 m/s (to the left)

Task: Find velocities after collision (assuming elastic collision).

#### Step-by-step Solution:

We use conservation of momentum and kinetic energy for elastic collisions.

Let:
- \( m_1 = 1 \, \text{kg}, v_{1i} = +2 \, \text{m/s} \)
- \( m_2 = 2 \, \text{kg}, v_{2i} = -1 \, \text{m/s} \)

Conservation of Momentum:
\[
m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}
\]
\[
(1)(2) + (2)(-1) = (1)v_{1f} + (2)v_{2f}
\]
\[
2 - 2 = v_{1f} + 2v_{2f}
\]
\[
0 = v_{1f} + 2v_{2f} \quad \text{(Equation 1)}
\]

Conservation of Kinetic Energy (elastic):
\[
\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2
\]
\[
(1)(4) + (2)(1) = (1)v_{1f}^2 + (2)v_{2f}^2
\]
\[
4 + 2 = v_{1f}^2 + 2v_{2f}^2
\]
\[
6 = v_{1f}^2 + 2v_{2f}^2 \quad \text{(Equation 2)}
\]

Now solve Equation 1 for \( v_{1f} \):
\[
v_{1f} = -2v_{2f}
\]

Substitute into Equation 2:
\[
6 = (-2v_{2f})^2 + 2v_{2f}^2 = 4v_{2f}^2 + 2v_{2f}^2 = 6v_{2f}^2
\]
\[
v_{2f}^2 = 1 \Rightarrow v_{2f} = \pm 1
\]

Try \( v_{2f} = 1 \): then \( v_{1f} = -2 \)

Try \( v_{2f} = -1 \): then \( v_{1f} = 2 \) → same as initial, not possible.

So, final velocities:
- \( v_{1f} = -2 \, \text{m/s} \) (left)
- \( v_{2f} = 1 \, \text{m/s} \) (right)

✔ Answer: After collision:
- 1 kg object moves left at 2 m/s
- 2 kg object moves right at 1 m/s

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Problem 2: Perfectly Inelastic Collision (Sticky Collision)

Given:
- Same masses and initial velocities as above.
- They stick together.

Task: Find final velocity.

Use conservation of momentum only (not kinetic energy).

\[
m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f
\]
\[
(1)(2) + (2)(-1) = (1+2) v_f
\]
\[
2 - 2 = 3 v_f
\]
\[
0 = 3 v_f \Rightarrow v_f = 0
\]

✔ Answer: Final velocity = 0 m/s (they stop).

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Problem 3: One object hits stationary object (inelastic or elastic?)

From your image, it seems:
- Object 1: 1 kg, 2 m/s → right
- Object 2: 2 kg, 0 m/s → stationary

After collision, they stick together.

Again, use conservation of momentum:

\[
m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f
\]
\[
(1)(2) + (2)(0) = (3) v_f
\]
\[
2 = 3 v_f \Rightarrow v_f = \frac{2}{3} \, \text{m/s}
\]

✔ Answer: Final velocity = \( \frac{2}{3} \, \text{m/s} \) to the right.

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Summary of Key Concepts:

| Type of Collision | Conservation Laws Used |
|-------------------|------------------------|
| Elastic | Momentum + Kinetic Energy |
| Inelastic | Momentum only (kinetic energy lost) |
| Perfectly Inelastic | Momentum only; objects stick together |

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