GCSE · Physics · Edexcel · Spec 1PH0

Newton's second law as F = change in momentum / time (impulse)

Stop a rolling trolley with one sharp shove or a long gentle push. Either way it ends up stopped, so why does one take so much more force?

Stopping a 20 kg trolley moving at 3 m/s: how hard, for how long?

13579020406080Time taken to stop, t (s)Size of resultant force, F (N)(5, 12)Force × time: 60 N s

Time taken to stop, t (s): 5. Size of resultant force, F (N): 12. Force × time: 60 N s

Drag the point along the curve. Watch the force as the stopping time grows, and keep an eye on force × time.

Watch out: Taking longer does not shrink the change in momentum. The trolley still goes from 3 m/s to rest, so its momentum changes by 60 kg m/s at every point on the curve. Only the force changes.

Newton's second law, the momentum way

F = (mv − mu) ÷ t

The resultant force equals the change in momentum divided by the time the change takes. In other words, force is the rate of change of momentum: how much momentum changes each second.

F = resultant force (N) · m = mass (kg) · u = initial velocity (m/s) · v = final velocity (m/s) · t = time taken (s). mv is the final momentum and mu is the initial momentum, so mv − mu is the change in momentum, in kg m/s.

One law, three ways to write it

F = m(v − u) ÷ t

Both momentum terms contain the mass, so pull it out as a common factor: mv − mu = m(v − u). Nothing has changed except how it is written.

1 / 4

Worked example: a ball that bounces back

Problem

A 0.5 kg ball hits a wall at 8 m/s and bounces straight back at 6 m/s. It is in contact with the wall for 0.05 s. Calculate the resultant force on the ball while it is touching the wall (treat the force as steady).

Spot the slip

Find the line where this answer goes wrong

A 0.16 kg ball travelling at 20 m/s is hit by a bat. It leaves the bat at 30 m/s in the opposite direction. The bat is in contact with the ball for 0.01 s. Calculate the force on the ball.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Force is how fast momentum changes, which is why the same change can take a huge force or a gentle one.

What you need to know

  • Newton's second law in momentum form: resultant force = change in momentum ÷ time taken for the change.
  • In symbols F = (mv − mu) ÷ t: F in N, m in kg, u (initial velocity) and v (final velocity) in m/s, t in s. mv − mu is the change in momentum in kg m/s.
  • This is the same law as F = m × a, because (v − u) ÷ t is the acceleration.
  • Impulse = force × time for which it acts (F × t). Impulse equals the change in momentum, and N s is equivalent to kg m/s.
  • For a given change in momentum, a longer time means a smaller force and a shorter time means a larger force.
  • In calculations, pick a positive direction; a velocity in the opposite direction is negative.

The big picture

Newton's second law can be written with momentum: the resultant force on an object equals its change in momentum divided by the time taken, F = (mv − mu) ÷ t. Take out the mass and it becomes F = m × a. Multiply by the time and you get impulse: force × time equals the change in momentum, and N s is the same unit as kg m/s. So for a given change in momentum, a longer time means a smaller force and a shorter time means a bigger one. In calculations, pick a positive direction and give any velocity the opposite way a minus sign.

Key points

1Force is the rate of change of momentum: how much the momentum changes each second.
2Change in momentum = final momentum − initial momentum = mv − mu.
3Rearranging gives F × t = mv − mu and t = (mv − mu) ÷ F.
4Same change in momentum, more time, less force. Same change, less time, more force.
5When something bounces back, its velocity changes sign, so its momentum changes by more than it would if it just stopped.

Worked example

Problem

A 40 kg sledge is pulled along with a resultant force of 120 N. Its velocity increases from 2 m/s to 8 m/s in the same direction. How long does this take?

⚠ Watch out

Forgetting the minus sign when something bounces back. If a ball arrives at 8 m/s and leaves at 6 m/s the other way, v is −6 m/s, not +6 m/s. Miss the sign and the change in momentum, and the force, come out far too small.

🧠

Memory hook

The change in momentum is the bill, and force × time is how you pay it. Pay it all at once with a big force in a short time, or bit by bit with a small force over a long time. The bill never changes.

✓

Check yourself

Without looking back: the same change in momentum now happens in a third of the time. What happens to the force, and why?

Flashcards

(13)
Newton's second law in terms of momentum: what does the resultant force equal?
The change in momentum divided by the time taken for that change (the rate of change of momentum).
Write the momentum form of Newton's second law in symbols.
F = (mv − mu) ÷ t
In F = (mv − mu) ÷ t, what do u and v stand for?
u is the initial velocity and v is the final velocity, both in m/s.
What does mv − mu represent, and what is its unit?
The change in momentum (final momentum minus initial momentum), in kg m/s.
Why is F = (mv − mu) ÷ t the same law as F = m × a?
mv − mu = m(v − u), and (v − u) ÷ t is the acceleration a, so F = m × a.
What is impulse?
The resultant force multiplied by the time for which it acts: F × t.
What is impulse equal to?
The change in momentum of the object: F × t = mv − mu.
Why is N s equivalent to kg m/s?
1 N = 1 kg m/s², so 1 N s = 1 kg m/s² × 1 s = 1 kg m/s.
The change in momentum is fixed. What happens to the force if the change takes longer?
The force gets smaller, because force × time must still equal the same change in momentum.
A change in momentum takes twice as long as before. How does the force compare?
It is half as big, because force = change in momentum ÷ time.
How do you handle a velocity in the opposite direction in a momentum calculation?
Choose one direction as positive first; a velocity the opposite way gets a negative sign.
Rearrange F = (mv − mu) ÷ t to find the time taken.
t = (mv − mu) ÷ F
Your answer for a force comes out negative. What does the minus sign tell you?
Its direction: the force acts opposite to the direction you chose as positive.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More Edexcel GCSE Physics topics

See the full Edexcel Physics curriculum →

How this lesson was checked. This Edexcel GCSE Physics (specification 1PH0)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 30 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.