Linear Momentum

Linear momentum is the product of a body’s mass and its linear velocity: $$\begin{equation}\begin{aligned} momentum(p)&=mass(m)\times velocity(v)\\ p&=mv\\ \end{aligned}\end{equation}$$

• The more mass the body has, the greater its momentum
• The faster the body is moving, the greater its momentum
• The SI unit is kilogram-metres per second ($kgms^{-1}$): $$\begin{equation}\begin{aligned} p=mv\rightarrow kg\times ms^{-1}\rightarrow kgms^{-1}\\ \end{aligned}\end{equation}$$
• Velocity is a vector hence momentum is also a vector
• Because momentum is a vector, when we work momentum problems, we must use a sign convention

Law of conservation of linear momentum

For an isolated system, the total momentum of the system before collision is equal to the total momentum of the system after collision.

For the collision between two bodies with masses $m_1$ and $m_2$: $$\begin{equation}\begin{aligned} total\ momentum\ before&=total\ momentum\ after\\ m_1v_1+m_2v_2&=m_1v_3+m_2v_4\\ \end{aligned}\end{equation}$$

Example 1

A stationary ball B mass $5\ kg$ is struck by another ball A of mass $2\ kg$ moving at $4\ ms^{-1}$. Find the final velocity of B if Ball A continues in the same direction as ball B but with velocity $1\ ms^{-1}$.

Example 2

Find the final velocity of B if Ball A is brought to rest by the collision.

Example 3

Find the final velocity of B if Ball A rebounds in the opposite direction with a velocity of $2\ ms^{-1}$.

Example 4

Ball C with mass $4\ kg$ moving left at $5\ ms^{-1}$ collides with ball D mass $2\ kg$ moving right at $8\ ms^{-1}$. What is the final velocity of C if D rebounds left with velocity $4\ ms^{-1}$?

Example 5

A bullet of mass $20\ g$ is fired from a gun with a muzzle velocity of $100\ ms^{-1}$. The mass of the gun is $5\ kg$. Find the recoil velocity of the gun.