The Principle of Conservation of Momentum
What is Momentum?
Before we can conserve something, we need to know what it is! In everyday language, momentum is like "oomph"—the property that keeps a moving object going. Scientifically, momentum is defined as the product of an object's mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction.
For example, a heavy truck moving at 10 m/s has much more momentum than a bicycle moving at the same speed because the truck has more mass. Similarly, the same truck moving at 20 m/s has twice the momentum it had at 10 m/s.
Stating the Conservation Principle
The Principle of Conservation of Momentum states that for a system of interacting objects (or particles), the total momentum remains constant, provided no external net force acts on the system. An "external force" is a force from outside the system. If the system is "closed" and "isolated" (meaning no external forces), the total momentum before an interaction equals the total momentum after.
This principle is not just a rule; it's a consequence of Newton's Third Law of Motion. When two objects interact, they exert equal and opposite forces on each other for the same amount of time. This means the impulse (change in momentum) on one object is equal and opposite to the impulse on the other, resulting in no net change in the total momentum of the system.
Different Types of Collisions
Collisions are the most common scenarios where we apply momentum conservation. However, not all collisions are the same. The table below summarizes the key types.
| Collision Type | Description | Is Kinetic Energy Conserved? | Example |
|---|---|---|---|
| Elastic | Objects bounce off each other perfectly. No permanent deformation or heat generation. | Yes | Two billiard balls colliding. |
| Inelastic | Objects stick together or deform. Some kinetic energy is lost as sound, heat, or deformation. | No | A car crash where the cars crumple. |
| Perfectly Inelastic | Objects stick together and move as one single object after the collision. | No (Maximum loss of kinetic energy) | A bullet embedding itself in a wooden block. |
It is vital to remember that momentum is conserved in all three types of collisions if the system is isolated. The conservation of kinetic energy, however, is what differentiates them.
Real-World Applications and Examples
The conservation of momentum isn't just a textbook idea; it's at work all around us. Let's explore some concrete examples.
1. Recoil of a Gun: When a bullet is fired from a gun, the gun exerts a forward force on the bullet. According to Newton's Third Law, the bullet exerts an equal and opposite force backward on the gun. Before firing, the total momentum of the system (gun + bullet) is zero. After firing, the bullet gains forward momentum, so the gun must gain an equal amount of momentum in the opposite direction to keep the total momentum at zero. This is why the gun "kicks" backward, an effect we call recoil.
2. Rocket Propulsion: How does a rocket move in the vacuum of space where there's nothing to push against? It uses momentum conservation. A rocket carries its own fuel. When the fuel is burned, hot gases are expelled backward at high speed from the engine. The backward momentum of the gas is accompanied by an equal forward momentum gained by the rocket, propelling it forward. This is an example of a system with changing mass, but the principle still holds.
3. Collisions in Sports: Imagine a baseball bat hitting a ball. The bat and the ball form a system. The force between them is internal. Although the bat slows down a little and the ball speeds up tremendously, the total momentum of the bat-ball system just before and just after the hit is the same (ignoring the force from the batter's hands for a moment). This principle is used to calculate the speed of the ball after it is hit.
Step-by-Step Problem Solving
Let's solve a classic perfectly inelastic collision problem to see the principle in action.
Scenario: A 1000 kg car moving at 20 m/s to the right collides with a stationary 1500 kg car. After the collision, the two cars lock together and move as one. What is their final velocity?
Step 1: Define the System and Direction. Our system is the two cars. We'll take motion to the right as positive.
Step 2: Write down the known values.
- Mass of first car, $ m_1 = 1000 $ kg
- Initial velocity of first car, $ v_{1i} = +20 $ m/s
- Mass of second car, $ m_2 = 1500 $ kg
- Initial velocity of second car, $ v_{2i} = 0 $ m/s
- Final velocity (both cars together), $ v_f = ? $
Step 3: Apply the Conservation of Momentum Formula.
Total momentum before = Total momentum after
$ m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f $
Step 4: Plug in the numbers and solve.
$ (1000 \times 20) + (1500 \times 0) = (1000 + 1500) v_f $
$ 20000 + 0 = 2500 v_f $
$ v_f = \frac{20000}{2500} = 8 $ m/s
Step 5: State the answer. The two cars move together to the right at 8 m/s.
Common Mistakes and Important Questions
Q: Is momentum conserved if there is friction?
A: It depends on how you define your system. If friction is an external force (e.g., friction from the road on a car), then the system's total momentum is not conserved. However, if you include the source of the friction (e.g., the Earth) in your system, then it can be considered an internal force, and momentum would be conserved, but this makes calculations very complex. For most practical problems, we consider our system as the interacting objects and treat friction as an external force that changes the total momentum.
Q: What is the difference between conserving momentum and conserving energy?
A: They are two separate and independent laws. Momentum is a vector quantity and is always conserved in an isolated system. Kinetic Energy is a scalar quantity and is not always conserved; it can be transformed into other forms like heat, sound, or potential energy (as in inelastic collisions). In an elastic collision, both are conserved. In an inelastic collision, only momentum is conserved.
Q: Can the conservation of momentum be applied in two dimensions?
A: Absolutely! Momentum is a vector, so the conservation principle applies separately in each direction (x and y). You would set up two separate equations: one for the conservation of momentum in the x-direction and one for the y-direction. This is essential for analyzing collisions that are not head-on, like two pool balls glancing off each other.
Footnote
1 SI unit: The International System of Units (Système International d'Unités), the modern form of the metric system and the world's most widely used system of measurement.
2 Vector Quantity: A physical quantity that has both magnitude and direction, such as force or velocity. It is typically represented by a bold symbol or an arrow above the symbol.
3 Newton's Third Law of Motion: States that for every action, there is an equal and opposite reaction. When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.