Apply the principle of the conservation of momentum to solve simple problems in one dimension

What is Momentum?

Momentum ($p$) is a measure of how hard it is to stop a moving object. It depends on two things:

• Mass ($m$) – heavier objects have more momentum.
• Velocity ($v$) – faster objects have more momentum.

Mathematically, momentum is written as:

$p = mv$

Think of a soccer ball: a heavier ball (more mass) or a faster kick (more velocity) makes it harder to stop.

Conservation of Momentum in One Dimension ⚖️

When two objects collide and no external forces act on them, the total momentum before the collision equals the total momentum after the collision.

$$\sum m_i v_{i,\text{initial}} = \sum m_i v_{i,\text{final}}$$

In one dimension, this means the direction of motion is along a straight line. If one object is moving left and the other right, we treat one direction as positive and the other as negative.

Example 1: Two Cars Colliding 🚗

Problem: Car A (mass $1.5\,\text{t}$) moves east at $20\,\text{m/s}$, Car B (mass $1.0\,\text{t}$) moves west at $15\,\text{m/s}$. They collide and stick together. Find their common speed after the collision.

1. Assign directions: east = +, west = –.
2. Calculate initial momenta:
   • Car A: $p_A = 1.5\times20 = 30\,\text{t·m/s}$ (positive)
   • Car B: $p_B = 1.0\times(-15) = -15\,\text{t·m/s}$ (negative)
3. Sum of initial momenta: $30 + (-15) = 15\,\text{t·m/s}$.
4. After collision, combined mass = $1.5+1.0 = 2.5\,\text{t}$.
5. Use conservation: $2.5\,v_{\text{final}} = 15$ → $v_{\text{final}} = 6\,\text{m/s}$ east.

Answer: The cars move together at $6\,\text{m/s}$ east.

Practice Problems 🧮

1. A $0.8\,\text{kg}$ ball moving at $10\,\text{m/s}$ collides with a $1.2\,\text{kg}$ ball at rest. If they stick together, what is their final speed?
2. Two ice skaters, each of mass $50\,\text{kg}$, push off from each other. One moves at $2\,\text{m/s}$ to the left. What is the speed of the other?
3. A $2\,\text{kg}$ projectile is fired at $5\,\text{m/s}$ into a $3\,\text{kg}$ block at rest. They collide elastically. Find the projectile’s speed after the collision.

Exam Tips 📚

• Always define a positive direction before starting.
• Check units: mass in kg, velocity in m/s, momentum in kg·m/s.
• Remember: elastic collision conserves both momentum and kinetic energy; inelastic collision conserves only momentum.
• When objects stick together, treat them as a single mass after the collision.
• Use a table to organise initial and final momenta for clarity.

Summary 📌

Momentum is the product of mass and velocity. In one-dimensional collisions, the total momentum before equals the total momentum after if no external forces act. By setting up equations and solving for unknowns, you can predict the outcome of many simple collision problems.

Keep practising, and remember: “Stop the moving object, stop the momentum.”