Momentum in A Level Math Paper 4 (Mechanics): common mistakes and examiner advice

Momentum is a small topic with high-value questions, usually a collision between particles, often combined with motion on a slope or an energy question. The reports describe the same errors again and again. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Not recognising a momentum problem. One report says the word "collision" should signal conservation of momentum, but only a minority used it. Others considered only the momentum after the collision, or only the loss in kinetic energy without conservation of momentum.
• Signs and direction. Momentum is a vector. Many sign errors come from forgetting this, and students also confused the velocity or mass of one particle with the other.
• The wrong mass. Using an incorrect mass for the total momentum after the collision (the most common wrong values were 6 or 2 rather than the correct 8 in one question), or using 1 for an unknown mass.
• The wrong velocity. Using the initial projection speed of a particle in place of its speed just before the collision, or using the speed at an earlier point (such as at B). Others used the speed found from momentum as the final answer for a different quantity.
• Only one of two possible speeds. In a question where the speed after collision could be positive or negative, students considered only +1 and not ±1, and found only one of the two possible speeds.
• Rejecting the impossible solution. Some students did not reject the algebraic solution that was inconsistent with the motion after the collision, or did not compare the subsequent speeds of two particles to decide whether a further collision would occur.
• Combined particle. A common error in a question where the particles join was to omit the time the combined particle takes to come back down the slope.
• Time and distance links. Using the same expression for the time for both particles, an incorrect gap between particles at the moment of collision, or sign errors in the equation for the time of collision.
• Using the wrong law. Applying Newton's second law to a car when an unknown braking force is acting, or to the system when the force in the tow-bar does not appear.
• Energy loss. Students found the loss for each particle but did not combine them, or found both losses but did not say which was greater.
• Given answers. Some students did not show enough working to show the given answer, such as v = 6.
• Notation. Using the same variable for the speeds of different particles is not good practice, and it contributes to mix-ups.

What examiners advise

• Write the conservation of momentum equation with a stated positive direction, and put in the velocities with signs.
• Draw a before-and-after diagram with a different letter for each speed.
• Use the correct speed just before the collision, which may need an earlier part of the question.
• Consider both possibilities if the question says a speed after a collision is given as a magnitude.
• Check the solution is physically possible: the particles must separate, and a trailing particle cannot move faster through the other.
• Show enough working to reach any printed answer.

Habits that help

• Write "momentum before = momentum after" and list each particle's mass and velocity underneath.
• Use different letters for the speeds of different particles.
• After solving, check whether the particles will collide again.

Questions students ask

How do I know when to use conservation of momentum?

The word "collision" should signal it, but only a minority of students used it. Write the equation with a stated positive direction and put the velocities in with signs, because momentum is a vector.

Which speed do I use for a particle that collides after moving?

The speed just before the collision, which may need an earlier part. Students used the initial projection speed or the speed at an earlier point such as B.

What if the speed after a collision could be positive or negative?

Consider both. In one question students considered only +1, not ±1, and found only one of two possible speeds.

How do I decide which solution to reject?

Check it is physically possible: the particles must separate, and a trailing particle cannot move faster through the other. Say when you reject a solution.

How do I avoid mixing up the speeds?

Draw a before-and-after diagram and give each speed a different letter. Using the same variable for the speeds of different particles contributes to mix-ups.