Here, we will look at how to solve momentum problems in both one and two dimensions using the law of conservation of linear momentum. According to this law, the total momentum of a system of particles remains constant as long as no external forces act on them. Therefore, solving momentum problems involve calculating the total momentum of a system before and after an interaction, and equating the two.

## How to Solve Momentum Problems

### 1D Momentum Problems

**Example 1**

*A ball with a mass of 0.75 kg travelling at a speed of 5.8 m s ^{-1} collides with another ball of mass 0.90 kg, also travelling in the same distance at a speed of 2.5 m s^{-1}. After the collision, the lighter ball travels at a speed of 3.0 m s^{-1} in the same direction. Find the velocity of the larger ball.*

According to the law of conservation of momentum, .

Taking the direction to the right on this digram to be positive,

Then,

**Example 2**

**An object of mass 0.32 kg traveling at a speed of 5 m s^{-1} collides with a stationary object having a mass of 0.90 kg. After the collision, the two particles stick and travel together. Find at which speed they travel.**

According to the law of conservation of momentum, .

Then,

**Example 3**

**A bullet having a mass of 0.015 kg is fired off a 2 kg gun. Immediately after firing, the bullet is travelling at a speed of 300 m s ^{-1}. Find the recoil speed of the gun, assuming the gun was stationary before firing the bullet.**

Let the recoil speed of the gun be . We will assume the bullet travels in the “positive” direction. The total momentum before firing the bullet is 0. Then,

.

We took the bullet’s direction to be positive. So, the negative sign indicates that the gun is travelling in the answer indicates that the gun is travelling in the opposite direction.

**Example 4: The ballistic Pendulum**

*The speed of a bullet from a gun can be found by firing a bullet at a suspended wooden block. The height* (**) that the block rises by can be measured. If the mass of the bullet (**

**) and the mass of the wooden block (**

**) are known, find an expression to calculate the speed**

**of the bullet.**From conservation of momentum, we have:

(where is the speed of the bullet+block immediately after collision)

From conservation of energy, we have:

.

Substituting this expression for in the first equation, we have

### 2D Momentum Problems

As mentioned in the article on the law of conservation of linear momentum, to solve momentum problems in 2 dimensions, one needs to consider momenta in and directions. Momentum will be conserved along each direction separately.

**Example 5**

*A ball of mass 0.40 kg, traveling at a speed of 2.40 m s ^{-1} along the *

*axis collides with another ball of mass 0.22 kg traveling**at a speed of mass 0.18, which is at rest. After the collision, the heavier ball travels with a speed of 1.50 m s*^{-1}with an angle 20^{o}to the

*axis, as shown below. Calculate the speed and direction of the other ball.***Example 6**

**Show that for an oblique collision (a “glancing blow”) when a body collides elastically with another body having the same mass at rest, the two bodies would move off at an angle of 90 ^{o} between them.**

Suppose the initial momentum of the moving body is . Take the momenta of the two bodies after the collision to be and . Since the momentum is conserved, we can draw up a vector triangle:

since , we can represent the same vector triangle with vectors , and . Since is a common factor to each side of the triangle, we can produce a similar triangle with just the velocities:

We know the collision is elastic. Then,

.

Canceling out the common factors, we get:

According to Pythagors’ theorem, then, . Since , so then . The angle between the two bodies’ velocities is indeed 90^{o}. This type of collision is common when playing billiards.