A staged problem follows one object through two or more events, and the energy at the end of one stage is the energy at the start of the next. Your job is to name the energy store at each stage and then choose the one formula that connects them.
This skill belongs to integrated physical reasoning in Co-ordinated Sciences. It returns whenever a question mentions a ramp, a fall, a braking vehicle or a swinging pendulum.
How do I turn a story into stages?
Read the question once and mark each place where something changes: released, reaches the bottom, enters rough ground, stops. Between two marks, ask: what energy store does the object have here, and where did it go next?
- List the stages in order.
- Name the energy at each: gravitational potential (mgh), kinetic (½mv²), or energy transferred to thermal stores by friction.
- Link neighbouring stages with “energy lost by one equals energy gained by the next”, unless the question says energy is wasted.
- Work out the unknown in the stage that asks for it, using the energy value carried across.
- Check units and size: mass in kg, height in m, speed in m/s, energy in J.
Worked example (invented data)
A 2.0 kg trolley is released from rest at a height of 0.80 m on a smooth ramp.
It then rolls onto a rough horizontal floor, where a constant friction force of 4.0 N stops it.
Use g = 10 N/kg. Find (a) the speed at the bottom, (b) the stopping distance.
Stage 1, release point. All the energy is gravitational potential energy.
GPE = mgh = 2.0 × 10 × 0.80 = 16 J
Stage 2, bottom of the ramp. The ramp is smooth, so all 16 J becomes kinetic energy.
½mv² = 16, so ½ × 2.0 × v² = 16, which gives v² = 16 and v = 4.0 m/s.
Stage 3, on the rough floor. The 16 J of kinetic energy is transferred to thermal energy by the work done against friction.
Work done = force × distance, so 4.0 × d = 16 and d = 4.0 m.
Re-check: 4.0 N × 4.0 m = 16 J, which matches the energy carried across. The speed check is v² = 2gh = 2 × 10 × 0.80 = 16, so v = 4.0 m/s again.
The mistake to watch for
A student writes mgh = mv² and loses the half from the kinetic energy formula.
Mistaken working: 16 = 2.0 × v², so v² = 8 and v = 2.8 m/s.
The kinetic energy formula is ½mv². Dropping the ½ makes the object appear to have only half its true speed squared.
The correction is to write the formula in full before substituting. A useful test is to ask whether the speed is sensible: a 0.80 m fall should give roughly 4 m/s, not under 3.
Check yourself
Use g = 10 N/kg. All data are invented.
1. A 0.50 kg ball is dropped from 1.8 m. Ignoring air resistance, what is its speed just before it hits the ground?
Show answer
GPE lost = 0.50 × 10 × 1.8 = 9.0 J. Then ½ × 0.50 × v² = 9.0, so 0.25 v² = 9.0 and v² = 36. v = 6.0 m/s.
2. A 1000 kg car travelling at 20 m/s brakes with a constant force of 5000 N. How far does it travel before stopping?
Show answer
KE = ½ × 1000 × 20² = 200 000 J. Work done = force × distance, so 5000 × d = 200 000 and d = 40 m. Check: 5000 × 40 = 200 000 J.
3. The 2.0 kg trolley from the worked example is released from the same height on a rough ramp, and reaches the bottom with 12 J of kinetic energy. How much energy was transferred to thermal stores, and what is its speed?
Show answer
Energy transferred = 16 − 12 = 4 J. For speed, ½ × 2.0 × v² = 12, so v² = 12 and v = 3.5 m/s (to 2 significant figures).
Where this leads next
Carry the same stage-by-stage habit into interpreting a wave observation with a numerical relationship. The scientific investigation critic helps you judge whether a measured speed from a ramp experiment is believable, and the integrated practice set mixes these stages with other topics.
If you can follow the example but freeze on a blank question, our teachers can look at how you read the story. That is the kind of work we do in online one-to-one Co-ordinated Sciences tuition.