Conservation of Momentum

Conservation of Momentum: A Fundamental Principle of Physics

1. Introduction to Conservation of Momentum

Definition: The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act upon it.

Mathematical Expression:

- Momentum before an interaction = Momentum after an interaction.

- If one body loses momentum, another body gains an equal amount of momentum.

2. Impulse-Momentum Theorem

Newton’s Second Law:

- F = \frac{\Delta p}{\Delta t}

- This indicates that force is proportional to the rate of change of momentum.

Impulse:

- Defined as the product of force and the time duration over which it acts:

- \text{Impulse} = F \cdot \Delta t = \Delta p = mv - mu

Importance of Follow-Through:

- In sports (e.g., golf, boxing), a longer contact time increases impulse, leading to greater changes in momentum and speed.

3. Justification of the Law of Conservation of Momentum

Collision Analysis:

- Consider two objects with masses M and m colliding.

- According to Newton’s Third Law, the forces they exert on each other are equal and opposite:

- F_{12} = -F_{21}

Change in Momentum:

- From Newton’s Second Law:

- \frac{\Delta p_1}{\Delta t} = -\frac{\Delta p_2}{\Delta t}

- Multiplying by time t :

- \Delta p_1 = -\Delta p_2

- This leads to:

- $\Delta p_1 + \Delta p_2 = 0

- Thus, the total change in momentum is zero, confirming that total momentum is conserved.

4. Collisions

Example: Road Traffic Accident:

- Before Collision:

- Momentum of blue car: m_b u_b

- Momentum of yellow car: m_y u_y

- Total momentum before:

m_b u_b + m_y u_y

- After Collision:

- Momentum of blue car: m_b v_b

- Momentum of yellow car: m_y v_y

- Total momentum after:

m_b v_b + m_y v_y

- Conservation Check:

- If total momentum before equals total momentum after, conservation holds.

5. Ballistic Pendulum

Setup: A bullet of mass m is fired into a stationary block of mass M .

Momentum Before:

- mv_0 (block is at rest)

Momentum After:

- Combined mass moves with speed v :

- (m + M)v

Conservation of Momentum:

- mv_0 = (m + M)v

Using Conservation of Energy:

- The kinetic energy of the block when the bullet hits it equals its potential energy at maximum height h :

- \frac{(m + M)v^2}{2} = (m + M)gh

- Solving for v :

- v = \sqrt{2gh}

Final Expression for Muzzle Velocity:

- Substituting v into the momentum equation gives:

- v_0 = \frac{(m + M) \sqrt{2gh}}{m}

6. Example Calculation

Typical Data:

- Mass of bullet m = 10g = 0.010kg

- Mass of block M = 1.3kg

- Height h = 1.45m

Calculation:

- v_0 = \frac{(1.310kg)(\sqrt{2 \cdot 9o.8m/s^2 \cdot 1.45m})}{0.010kg}

- Result:

- v_0 \approx 698 m/s