Momentum & Force-Time Graphs
What Is Momentum?
Momentum is a property of any moving object. It depends on both the mass and velocity of the object. The equation for momentum is:
$$p = m \times v$$
where \(p\) is momentum in kg m/s, \(m\) is mass in kg, and \(v\) is velocity in m/s. Because velocity is a vector, momentum is also a vector quantity — it has both magnitude and direction.
Momentum = mass × velocity. A 50 kg skater moving at 4 m/s has a momentum of 200 kg m/s. Doubling the velocity doubles the momentum.
Conservation of Momentum
In a closed system (one where no external forces act), the total momentum before an event is equal to the total momentum after it. This is the conservation of momentum.
This principle applies to all collisions and explosions. For a collision between two objects:
$$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$$
In a perfectly inelastic collision, the two objects stick together and move as one after the collision. For example, a 2 kg trolley moving at 3 m/s collides with a stationary 1 kg trolley. They stick together:
Total momentum before = \(2 \times 3 + 1 \times 0 = 6\) kg m/s
Total momentum after = \((2 + 1) \times v = 6\), so \(v = 2\) m/s
In an explosion, the total momentum before is zero (everything is stationary). After the explosion, the momentum of each part must add to zero — the parts move in opposite directions.
Newton’s Second Law and Momentum
Newton’s second law can be written in terms of momentum. The resultant force on an object equals the rate of change of momentum:
$$F = \frac{mv - mu}{t}$$
where \(F\) is the force in newtons (N), \(mv - mu\) is the change in momentum in kg m/s, and \(t\) is the time in seconds (s). This form of the equation is particularly useful for understanding collisions and force-time graphs.
The area under a force-time graph equals the change in momentum (impulse). A larger force applied for a shorter time produces the same change in momentum as a smaller force applied for a longer time.
Stopping Distance
The stopping distance of a vehicle is the sum of the thinking distance and the braking distance:
$$\text{stopping distance} = \text{thinking distance} + \text{braking distance}$$
Thinking distance is affected by the driver’s reaction time. Factors that increase reaction time include tiredness, alcohol, drugs, distractions (such as using a mobile phone), and poor visibility. A typical reaction time is 0.2–0.9 seconds.
Braking distance is affected by: the speed of the vehicle (higher speed means longer braking distance), the mass of the vehicle, the condition of the brakes, the condition of the road surface, and the amount of friction between the tyres and the road.
Increasing speed has a dramatic effect on stopping distance because both thinking distance and braking distance increase with speed. Braking distance increases with the square of the speed.
Large Decelerations and Safety
When a vehicle stops very quickly, the deceleration is very large. Using \(F = \frac{mv - mu}{t}\), if the time is very short, the force is very large. This can cause serious injuries to passengers.
Crumple zones, seat belts, and airbags all work on the same principle: they increase the time over which the momentum changes, which reduces the force experienced by the passengers. This is a direct application of the equation \(F = \frac{\Delta p}{t}\) — increasing \(t\) decreases \(F\) for the same change in momentum.
Safety features such as crumple zones, seat belts, and airbags increase the time of deceleration, reducing the force on passengers for the same change in momentum.
In this core practical, you vary the masses added to trolleys to investigate how force, mass, and acceleration are related. A trolley is accelerated along a track by a hanging mass. Light gates measure the acceleration.
The independent variable is the force (changed by varying the hanging mass). The dependent variable is the acceleration. The control variables include the total mass of the system. As force increases, acceleration increases proportionally (for constant mass), confirming \(F = ma\).
Exam Tip: For momentum calculations, always assign a direction as positive and stick with it. Objects moving in opposite directions have opposite signs for velocity. When using \(F = \frac{mv - mu}{t}\), remember to subtract the initial momentum from the final momentum, not the other way round.
Momentum is mass multiplied by velocity and is always conserved in a closed system. Newton’s second law links force to the rate of change of momentum, and this explains why safety features like crumple zones reduce the force in a collision by increasing the stopping time. Stopping distance depends on both thinking distance and braking distance, each affected by different factors.