Momentum
Start here
Lesson overview
Momentum as mass × velocity, Newton's first and third laws, impulse as force × time, and why stopping slowly reduces the force.
Go to lesson quiz →Exam guidance · JAMB UTME Speed Lens
Core idea (Kinematics,SUVAT & Newton's Laws of Motion): Pick a positive direction first, list knowns/unknowns (u, v, a, s, t), draw the free-body diagram before writing ΣF = ma, and check vector signs.
Pacing
~40 seconds per MCQ (180 questions in 120 minutes, no negative marking).
How JAMB marks this
Fast dimensional/sign elimination and standard formula shortcuts — skip multi-line algebra when substitution tests options faster.
Watch out for
UTME distractors routinely include the intermediate value (e.g., forgetting to convert cm³ to dm³ or °C to Kelvin).
Lesson content
Overview
A loaded lorry and a bicycle may both be moving at 10 m/s along the Lagos–Ibadan expressway, but everybody knows which one is harder to stop. The quantity that captures this "quantity of motion" is momentum. It depends on how much mass is moving and how fast it is moving. In this lesson you will calculate momentum, connect it to Newton's first and third laws, and use impulse (force × time) to explain why catching a ball with soft hands, crumple zones and seat belts all reduce the force on a body. UTME questions on momentum range from one-line calculations of p = mv to force–time graphs and rebounds, where the direction of motion decides the answer.
Key concept: momentum, inertia and impulse
Linear momentum is the product of the mass of a body and its velocity: p = mv. It is a vector, so it has the same direction as the velocity. Its SI unit is the kilogram metre per second (kg m/s), which is the same as the newton second (N s). Inertia is the reluctance of a body to change its state of rest or of uniform motion. Mass is the measure of inertia: the bigger the mass, the harder it is to start, stop or turn the body. Impulse is the product of a force and the time for which it acts: impulse = Ft. Impulse equals the change in momentum it produces. The principle of conservation of linear momentum: when no external force acts on a system of bodies, their total momentum stays constant.
Newton's first and third laws, and momentum
Newton's first law: a body stays at rest, or keeps moving with uniform velocity in a straight line, unless a resultant external force acts on it. This is why passengers in a danfo jerk forward when the driver brakes suddenly: their bodies tend to keep moving at the old velocity, and only the seat belt (or the seat in front) supplies the force to stop them. Newton's second law, in its fuller form, says the resultant force equals the rate of change of momentum: F = (mv − mu) ÷ t. For a fixed mass this becomes F = ma, the form used in the companion lesson on the second law. Newton's third law: whenever body A exerts a force on body B, B exerts an equal and opposite force on A. The two forces act on different bodies, so they never cancel each other. When a gun fires a bullet forward, the bullet pushes the gun backwards with the same size of force for the same time, so the gun and the bullet receive equal and opposite momenta. That backward kick is called recoil. Because momentum is a vector, choose one direction as positive before you calculate. A ball moving left at 5 m/s has velocity −5 m/s if right is positive. Forgetting this sign is the single most common cause of wrong answers in rebound questions. Rearranging F = Δp ÷ t gives Δp = Ft. For a given change in momentum, the longer the force acts, the smaller the force. Goalkeepers pull their hands back when catching a hard shot, and cars have crumple zones and airbags, for exactly this reason.
Formulae to know
Momentum: p = mv (kg m/s or N s) Newton's second law: F = (mv − mu) ÷ t = change in momentum ÷ time Impulse: Ft = mv − mu (the change in momentum) Impulse from a force–time graph = area under the graph Conservation of momentum: total momentum before = total momentum after (no external force) Unit conversion: km/h ÷ 3.6 = m/s
Worked example 1: momentum of a car
A car of mass 1,200 kg travels at 72 km/h. Find its momentum. What was its momentum earlier, when it was moving at 15 m/s?
Active Worked Example · Predict Step 1
4 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 2: impulse from a force–time graph
A footballer kicks a stationary ball of mass 0.45 kg. The force on the ball rises steadily from 0 to 900 N and falls back to 0 in a total contact time of 0.01 s, as shown in the graph below. Find the impulse, the speed of the ball and the average force.
Active Worked Example · Predict Step 1
4 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 3 (UTME standard): a rebound
A ball of mass 0.2 kg hits a wall at 15 m/s and rebounds along the same line at 10 m/s. The ball is in contact with the wall for 0.05 s. Find the change in momentum of the ball and the average force exerted on it by the wall.
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Common traps
• Momentum is a vector. In a rebound, the change in velocity is the sum of the two speeds, not the difference. • Convert km/h to m/s (÷ 3.6) and grams to kilograms before multiplying. • Kg m/s and N s are the same unit. Do not reject a correct option because it uses the other one. • Action and reaction act on different bodies, so they never cancel. A horse can pull a cart because the forces on the cart are what matter for the cart. • A body at rest has zero momentum, not "no value". Include it as zero when adding momenta. • Impulse is the area under a force–time graph, not the gradient.
Summary
• Momentum p = mv is a vector measured in kg m/s (= N s). • Newton's first law describes inertia; mass measures inertia. • Newton's second law: F = rate of change of momentum = (mv − mu) ÷ t. • Newton's third law: action and reaction are equal, opposite and act on different bodies; this explains recoil. • Impulse Ft = change in momentum = area under a force–time graph. A longer stopping time means a smaller force.
Mastery Check · Lesson Quiz · pass 60%
Sign in to take the quiz and track your progress.
Create a free account to take quizzes, earn checkpoints and track your scores — no card needed.
Create your free account