How Do You Calculate Momentum in a Collision? (HL)

Calculate each object's momentum using mass multiplied by velocity, then apply the conservation of linear momentum: in an isolated system, total momentum before a collision equals total momentum after it.

Momentum is a vector quantity:

$$p = mv$$

where $p$ is momentum in $\text{kg m s}^{-1}$, $m$ is mass in $\text{kg}$, and $v$ is velocity in $\text{m s}^{-1}$. Because velocity has direction, choose one direction as positive and use negative velocities for motion in the opposite direction.

For two colliding bodies:

$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$

Here, $u$ represents initial velocity and $v$ represents final velocity. This equation applies when the system experiences no significant net external impulse during the collision.

During the short collision, the players exert equal and opposite forces on one another for the same time interval. Their impulses are therefore equal in magnitude and opposite in direction, redistributing momentum between the players while leaving the system total unchanged. External forces such as friction are usually negligible over this brief interval.

Worked SEHS example

A $70\text{ kg}$ rugby player runs right at $5\text{ m s}^{-1}$ and tackles a stationary $80\text{ kg}$ player. They move together after impact.

Initial total momentum:

$$p_{\text{initial}} = (70)(5) + (80)(0) = 350\text{ kg m s}^{-1}$$

Because they move together, their combined mass is $150\text{ kg}$:

$$350 = (70 + 80)v$$ $$v = \frac{350}{150} = 2.33\text{ m s}^{-1}$$

Both players therefore move right at $2.33\text{ m s}^{-1}$ immediately after the collision.

Check the answer by substituting the final velocity: $(150)(2.33) \approx 350\text{ kg m s}^{-1}$, allowing for rounding. The positive result confirms rightward motion.

A common misconception is that momentum is conserved for each object separately. It is the total momentum of the system that is conserved.

In an IB SEHS response, state the positive direction, show the momentum equation, include signs and units, and interpret the final velocity's direction. Link the calculation to Newton's laws of motion, particularly equal and opposite interaction forces during impact.