Chapter 1 · Mechanics
1-9 Momentum and Mechanical Energy
❓ Lesson question
Two trolleys collide, stick together and move more slowly. If their total momentum is conserved, why does their total kinetic energy decrease, and to what forms of energy is it transferred?
You will learn to
- 1State when total momentum is conserved, and distinguish between elastic and inelastic collisions.
- 2Calculate the decrease in total kinetic energy during an inelastic collision.
- 3Explain how kinetic energy is transferred to other forms during an inelastic collision.
A moving laboratory trolley collides with a second trolley that is initially at rest. Adhesive pads on their ends cause the trolleys to stick together and move with a common velocity after the collision. Their common speed is lower than the initial speed of the moving trolley. A short sound may be heard during the collision, and the trolleys and pads may undergo slight deformation. These observations suggest that some kinetic energy is transferred to other forms of energy.
✍️ Predict before you continue
Write your answer before reading on. A prediction you got wrong and then examined teaches you more than an answer you were handed.
(a) After they stick together, is the total momentum smaller, the same, or larger than before?
(b) What happens to the total kinetic energy — does it become smaller, stay the same, or become larger?
🔬 Explore — Momentum and kinetic energy in an inelastic collision
What you’ll do: Record a slow-motion video of two trolleys colliding, then compare their motion before and after the collision.
- 1Measure and record the mass of each trolley. Place trolley B at rest on a level, low-friction track.
- 2Release trolley A so that it collides with trolley B. Hook-and-loop pads on the facing ends should cause the trolleys to stick together. Record the entire collision using slow-motion video.
- 3Place a measuring tape parallel to the track and close to the path of the trolleys. Use the video frame rate and the distance travelled over several frames to determine the speed of trolley A immediately before the collision, and the common speed of the joined trolleys immediately after.
- 4Note whether sound, vibration or visible deformation occurs during the collision.
- 5Add a mass securely to trolley B and repeat the procedure. Keep the initial speed of trolley A as nearly the same as possible.
Analyse what you collected. (a) For each trial, calculate the total momentum and total kinetic energy before and after the collision. Is each approximately conserved? (b) Explain how the observations of sound, vibration or deformation provide evidence that some kinetic energy was transferred to other forms of energy.
In Explore, the two trolleys moved together with a common speed lower than the initial speed of the moving trolley. Using the measured masses and speeds shows a more precise result: the total momentum before and after the collision was approximately the same, but the total kinetic energy after the collision was smaller.
The kinetic energy was not destroyed. Some of it was transferred to other forms of energy, including internal energy associated with deformation and vibration, and sound.
🎛️ Simulation — figure 1-9-3 with a dial on it
0 ≤ e < 1 → inelastic. Mechanical energy decreases.
Total momentum (a vector — the halves can cancel)
Drag e across its whole range. These two never differ. Momentum is conserved for every value of e, elastic or not.
Total kinetic energy (has v² — nothing can cancel)
These match only at e = 1. Everywhere else the second bar is shorter, and the missing part has become internal energy and sound.
Final velocity of A
1.00 m/s
Final velocity of B
2.00 m/s
Kinetic energy lost
4.50 J
to deformation, vibration and sound
Press Worked example p.53. The book prints v′A = 1.0 m/s, v′B = 2.0 m/s and a loss of 4.5 J; the boxes above must read the same. Then drag e up to 1 without touching anything else: the green pair closes to zero loss, and the indigo pair does not move at all. That single motion is the answer to this lesson's question — momentum does not care how bouncy the collision was, and kinetic energy does.
Point
- • The sum of a system's kinetic energy and potential energy is called mechanical energy. For mass m at speed v,Ek=21mv2.
- • When objects collide, their relative speed after the collision is less than or equal to their relative speed before it, so for ordinary (non-explosive) collisions the coefficient of restitution e — the ratio of the relative speed after the collision to the relative speed before it — satisfies 0≤e≤1.
- • When e=1 it is a perfectly elastic collision; when0≤e<1 an inelastic collision; and the inelastic collision withe=0 is called a perfectly inelastic collision — the bodies become one body after the collision.
- • In an elastic collision (e=1) mechanical energy is conserved. In an inelastic collision (0≤e<1) mechanical energy decreases: some of it is transferred to internal energy associated with deformation and vibration, and to sound.
📐 Equation box
Kinetic energy of a body
Conservation of momentum (isolated system)
Coefficient of restitution
Kinetic energy lost
🛠️ Worked example (book p.53)
Situation
A small ball A of mass 1.0 kg moves to the right at 4.0 m/s and collides head-on with a stationary small ball B of mass 1.5 kg. The coefficient of restitution is 0.25. (a) Determine the final velocities of the two balls after the collision in m/s. (b) Determine the mechanical energy lost in the collision in J. Take the rightward direction as positive.
Answer. Taking rightward as positive, p=4.0(6.0)=24 kg·m/s, sov′=24÷7.0=3.43 m/s.Ek before =21(4.0)(6.0)2=72 J; after =21(7.0)(3.43)2=41.1 J. The loss is 30.9 J — well over a third of the original energy, which is typical of a perfectly inelastic collision between comparable masses.
In a New Context
The crumple zone. A vehicle collision is largely inelastic. During the impact, the crumple zone deforms, so the vehicle rebounds less and its kinetic energy decreases. The decrease in kinetic energy is transformed mainly into internal energy through permanent deformation, internal vibrations and heating, while a smaller amount is transferred to the surroundings as sound. The deformation also increases the time over which the vehicle's momentum changes, reducing the average force acting on the driver and passengers.
Your turn: Explain why a car designed to crumple in a crash protects its passengers better than one built to remain rigid.
Answer. For two reasons that work together, and both come from earlier lessons. First, from 1-7: the passenger's momentum must change by the same amount whatever the car is made of, and FΔt=Δp, so stretching the collision from milliseconds to tenths of a second cuts the average force by the same factor. Second, from this lesson: crumpling makes the collision more inelastic, so more kinetic energy is spent deforming metal and less is left to throw the car back off the obstacle — and a rebound would mean a larger momentum change, not a smaller one. A rigid car does neither: it stops in a very short time and it bounces.
🧪 Nature of Science — Evidence — where the 'lost' kinetic energy goes
In 1850, James Joule published his paddle-wheel experiment in the Philosophical Transactions. Falling weights turned a paddle that stirred water, and the mechanical work produced a measurable rise in the water's temperature. This was evidence that the work was not lost but converted into internal (thermal) energy. The kinetic energy "lost" in an inelastic collision changes in the same way: it becomes internal energy, which may be measured as a temperature rise.
An answer to "your turn". Because "lost" is not a measurement — it is the name we give to a gap in our accounts, and it would sound the same whether the energy had gone somewhere or ceased to exist. A temperature rise is a positive observation with a number attached: it says where the energy went, and it lets the two figures be compared. If the measured thermal energy matched the missing kinetic energy, conservation of energy survives; if it did not, we would know the account was still incomplete. This is the same standard Pauli met in the previous lesson — a claim that can be checked and could have failed is worth more than one that cannot.
Your turn: Suggest why a measured temperature rise is stronger evidence than simply stating that the energy is lost.
✅ Lesson question, answered
Two trolleys collide, stick together and move more slowly. If their total momentum is conserved, why does their total kinetic energy decrease, and to what forms of energy is it transferred?
The total momentum is conserved during the collision. However, the decrease in the total kinetic energy of the colliding bodies is converted mainly into internal energythrough deformation, internal vibrations and heating, while a smaller fraction is carried away by sound waves.
There is no contradiction between the two statements. Momentum is conserved because the forces the trolleys exert on each other are equal, opposite and act for the same time, so the impulses cancel — regardless of what the trolleys are made of. Kinetic energy carries no such guarantee: the contact forces do work in deforming the pads, and that work leaves the mechanical account entirely.
- mechanical energy
- — The sum of a system's kinetic energy and potential energy.
- coefficient of restitution
- — The ratio of the relative speed after a collision to the relative speed before it.
- elastic collision
- — A collision with e = 1, in which kinetic energy is conserved.
- inelastic collision
- — A collision with 0 ≤ e < 1, in which mechanical energy decreases.
- perfectly inelastic collision
- — The case e = 0: the bodies move off together as one body.
- internal energy
- — Energy associated with deformation, vibration and heating of the bodies themselves.
How the real paper is built
The ministry sets 50% multiple choice and 50% written answer. They test different things: the MCQ half rewards eliminating fast and spotting a distractor built from a plausible wrong method; the written half asks you to show that you understand why. Practise both halves — being fluent at one and weak at the other costs you exactly half the paper.
One step. You are given the formula; put the numbers in.
Two steps, or you must choose the right component before you can start.
Multi-step, or a case the textbook does not set and examiners do.
📝 Exam-style question
Sign In to Keep Going
You've reached the end of the free preview. Sign in — and upgrade to Plus or Pro — to keep going with Physics.
Full Access
Unlock all 12 lessons