Forces and Momentum Explained
Why does a heavy truck take longer to stop than a bicycle travelling at the same speed? Why does a gun recoil when it fires? The concept of momentum — mass in motion — answers both questions, and the principle of its conservation is one of the most useful tools in physics.
Defining Momentum
Momentum (p) is defined as the product of an object's mass and its velocity:
Mass is measured in kilograms (kg), velocity in metres per second (m/s), so momentum has units of kg m/s. Because velocity is a vector — it has both magnitude and direction — momentum is also a vector. A 1,500 kg car travelling north at 20 m/s has momentum of 30,000 kg m/s directed north. The same car moving south has momentum of the same magnitude but in the opposite direction.
This directionality matters in calculations. When two objects collide head-on, you must assign one direction as positive and treat velocities in the opposite direction as negative before you apply any formula.
Balanced and Unbalanced Forces
A force is any interaction that can change an object's state of motion. Forces are also vectors, and the net force on an object is the vector sum of all forces acting on it.
When forces are balanced (net force = 0), an object either stays still or continues moving at constant velocity. A book resting on a table has gravity pulling it down and the normal force from the table pushing it up. These are equal and opposite, so the book does not accelerate.
When forces are unbalanced, there is a non-zero net force, and the object accelerates in the direction of that net force. This is simply Newton's second law: F = ma, or equivalently, net force = rate of change of momentum.
Newton originally stated his second law in terms of momentum: force equals the rate of change of momentum. In symbols: F = Δp / Δt. This form is actually more general than F = ma, because it still works even if the object's mass changes (as in a rocket burning fuel).
Impulse
Rearranging F = Δp / Δt gives:
The product of force and the time over which it acts is called impulse, measured in newton-seconds (N s), which is equivalent to kg m/s. Impulse equals the change in momentum produced.
This relationship has practical consequences. If you need to change an object's momentum by a fixed amount, you can do it with a large force over a short time — or a smaller force over a longer time. Airbags exploit this: in a collision, the car decelerates rapidly, but the airbag extends the time over which the force acts on the passenger's head, reducing the peak force and the risk of injury. A similar logic explains why padded surfaces are safer than hard ones, and why gymnasts bend their knees when landing rather than keeping their legs rigid.
Conservation of Momentum
The law of conservation of momentum states: in a closed system with no external net forces, the total momentum before an event equals the total momentum after it.
This law follows directly from Newton's third law. When two objects interact, they exert equal and opposite forces on each other for equal times, so they exchange equal and opposite impulses. One gains exactly as much momentum as the other loses, and the total stays constant.
A worked example: a 2 kg ball moving at 5 m/s east collides with a stationary 3 kg ball. After the collision the 2 kg ball moves at 1 m/s east. What is the velocity of the 3 kg ball?
- Total momentum before: (2 × 5) + (3 × 0) = 10 kg m/s
- Momentum of 2 kg ball after: 2 × 1 = 2 kg m/s
- Momentum of 3 kg ball after: 10 − 2 = 8 kg m/s
- Velocity of 3 kg ball: 8 ÷ 3 = 2.67 m/s east
Elastic and Inelastic Collisions
Conservation of momentum applies to all collisions, but kinetic energy is not always conserved. This distinction defines two types of collision.
In a perfectly elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other with no loss of energy to heat, sound, or deformation. Collisions between gas molecules are approximately elastic. Billiard balls are a common classroom example, though real billiard ball collisions lose a small amount of energy.
In a perfectly inelastic collision, the objects stick together and move as one after the collision. Momentum is still conserved, but kinetic energy is not — some is converted to other forms. A lump of clay hitting a stationary clay target, or two railway carriages coupling together, approximates this case. Most real collisions fall somewhere between these extremes.
To find the final velocity of two objects that stick together:
Solving for v𝒓 gives the common final velocity.
Momentum in Two Dimensions
In problems where objects collide at angles, momentum must be conserved separately in each direction. You resolve each object's velocity into horizontal (x) and vertical (y) components, apply conservation in both directions independently, and then combine the components to find the resultant velocity.
A snooker ball struck off-centre is a familiar two-dimensional collision. The cue ball does not travel straight after impact — some of its forward momentum is transferred to the target ball, and both emerge at angles. By treating horizontal and vertical momentum independently, you can predict the paths of both balls.
Summary
Momentum is mass times velocity, and its direction matters. An unbalanced net force changes an object's momentum at a rate equal to the force applied; the product of force and time is impulse. In any closed system, total momentum is conserved — this principle underlies everything from airbag design to rocket propulsion. Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve only momentum. In two-dimensional problems, momentum conservation is applied to each component direction separately.