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C2.1 · Use work, energy, impulse, momentum, and collision terminology

Learn to use work, energy, impulse, momentum, and collision terminology through clear examples and targeted practice.

Ontario Grade 12 Physics

Energy and Momentum

A vocabulary and relationship guide for SPH4U expectation C2.1

In SPH3U, you studied motion, forces, and energy. This lesson builds on those ideas by giving precise names to how objects move and interact. A physical system is the object or group of objects being studied. The reference frame is the viewpoint used to describe positions and motion; here, use the ground as the frame unless a problem says otherwise. Choose a positive direction before assigning signs. In one-dimensional examples, rightward is positive. A scalar has magnitude only, while a vector has both magnitude and direction. Work and energy are scalars. Momentum, impulse, force, and velocity are vectors. The sign of a vector component tells you its direction relative to the chosen positive direction.

What you will learn

Work and energy describe transfer and motion

A force is a push or pull. Displacement is the change in an object's position, including direction. In physics, work describes energy transferred when a force acts through a displacement. A force perpendicular to the displacement transfers no energy by work in this model. When force and displacement point in the same direction, the work is positive; when they point in opposite directions, the work is negative.
Work is a scalar, even though force and displacement are vectors. Its SI unit is the joule, abbreviated J. One joule is equivalent to one newton metre. The equation uses the component of force along the displacement, so it also applies when the angle between them is not zero.
Energy is a scalar quantity associated with an object's ability to cause change or transfer energy. Kinetic energy is the energy of motion. For an object of mass mm moving at speed vv, kinetic energy depends on both mass and speed. Speed is a scalar; velocity is a vector. The work-energy relationship says that the net work on an object equals its change in kinetic energy. Net work means the combined work done by all forces on the object.
A positive change in kinetic energy means the object ends with more kinetic energy than it started with. A negative change means it ends with less. Work and energy use the same SI unit, the joule. This relationship concerns the object's initial and final kinetic energies; it does not say that every individual force does positive work.
W=Fdcos⁡θ=ΔEkW=Fd\cos\theta=\Delta E_k

Momentum and impulse describe motion and its change

Momentum describes an object's motion in a way that depends on both its mass and velocity. Because velocity has direction, momentum is a vector. Its SI unit is kilogram metre per second. In one dimension, a positive or negative momentum value indicates direction relative to the chosen axis.
Impulse describes the effect of a force acting over a time interval. For a constant net force, impulse is the net force multiplied by the elapsed time. Impulse is a vector, and its SI unit is the newton second. A newton second is equivalent to a kilogram metre per second, so impulse and momentum can be compared directly.
The impulse-momentum relationship states that the net impulse on an object equals its change in momentum. A force acting for a longer time can produce a larger impulse if the force is unchanged. A force in the opposite direction to the object's motion can produce a negative impulse and reduce its momentum in that direction.
When studying an interaction, state what is included in the system. For two objects treated together, forces they exert on each other are internal to that system. If the net external force on the system is zero, its total momentum remains constant. This condition is called conservation of momentum. If a net external force acts, do not assume the system's momentum is unchanged.
p⃗=mv⃗,J⃗=F⃗netΔt=Δp⃗\vec p=m\vec v,\qquad \vec J=\vec F_{\mathrm{net}}\Delta t=\Delta\vec p

Collision terminology and system choices

A collision is an interaction between objects during which they exert forces on each other over a time interval. The word collision does not require objects to be hard or to make a sound. The system might be one object, or it might include all objects involved. The system choice matters because it determines which forces count as external.
In an isolated collision model, the net external force on the chosen system is zero during the interaction. The system's total momentum before the collision then equals its total momentum after it. In a one-dimensional problem, use signed velocity components. Add the momenta with their signs; do not treat every momentum as positive.
An elastic collision is one in which the system's total kinetic energy before and after the collision is the same. An inelastic collision is one in which that total kinetic energy changes. Momentum can still be conserved in an inelastic collision if the system has no net external force. In a perfectly inelastic collision, the objects remain together after impact and share a final velocity.
A collision label describes what happens to the system's kinetic energy or to the objects after contact. It does not by itself tell you whether momentum is conserved. Check the external forces and the chosen system separately.
∑ip⃗i, before=∑ip⃗i, after\sum_i \vec p_{i,\,\mathrm{before}}=\sum_i \vec p_{i,\,\mathrm{after}}

Reading relationships with units and direction

Before calculating, name the system, frame, positive direction, known values, and unknown. Then select a relationship that matches the situation. Use joules for work and energy, kilogram metres per second for momentum, and newton seconds for impulse. These units help you notice when a quantity has been mislabeled.
For vector quantities in one dimension, include signs in substitutions and final answers. A negative velocity or momentum does not mean the quantity is physically smaller than zero; it indicates that the direction is opposite to the chosen positive direction. A scalar such as kinetic energy cannot be negative.
After calculating, check the units and direction. Also ask whether the result fits the situation. For example, if two objects stick together, their shared speed should follow from the total initial momentum and their combined mass, not from adding their speeds.

Worked example

Work by a force along a displacement

A crate is the system. In the ground frame, it moves 4.0 m to the right while a constant 18 N force acts to the right. Right is positive. Find the work done by this force.
  1. Identify the angle
    The force and displacement point in the same direction, so the angle between them is zero. The requested quantity is scalar work.
    θ=0∘\theta=0^\circ
  2. Apply the work relationship
    Work equals the force magnitude multiplied by displacement and by the cosine of the angle between them. Use the given SI values.
    W=(18 N)(4.0 m)cos⁡(0∘)=72 JW=(18\,\mathrm{N})(4.0\,\mathrm{m})\cos(0^\circ)=72\,\mathrm{J}
  3. Check the result
    A newton metre is a joule. The positive result fits a force acting in the direction of motion. Two significant figures are appropriate for the given values.
    1 N m=1 J1\,\mathrm{N\,m}=1\,\mathrm{J}
Answer: The force does 72 J72\,\mathrm{J} of work on the crate.
Check: The units reduce to joules, and the sign is positive because force and displacement point in the same direction.

Worked example

Impulse changes momentum

A 0.50 kg cart is the system. In the ground frame, right is positive. A constant net force of 6.0 N acts to the left for 0.25 s. Find the impulse and the change in the cart's momentum.
  1. Assign direction
    The force is leftward, opposite the positive direction, so its signed value is negative. The time interval is positive.
    Fnet=−6.0 NF_{\mathrm{net}}=-6.0\,\mathrm{N}
  2. Calculate impulse
    For a constant net force, impulse is net force multiplied by elapsed time. Its direction follows the force.
    J=(−6.0 N)(0.25 s)=−1.5 N sJ=(-6.0\,\mathrm{N})(0.25\,\mathrm{s})=-1.5\,\mathrm{N\,s}
  3. Connect impulse to momentum change
    The impulse-momentum relationship makes the change in momentum equal to the net impulse. The negative sign means the change points left.
    Δp=J=−1.5 kg m/s\Delta p=J=-1.5\,\mathrm{kg\,m/s}
Answer: The impulse is 1.5 N s1.5\,\mathrm{N\,s} leftward, and the cart's momentum changes by 1.5 kg m/s1.5\,\mathrm{kg\,m/s} leftward.
Check: A newton second is equivalent to a kilogram metre per second. The leftward direction agrees with the applied net force.

Worked example

Objects that stick together

Two carts form the system in the ground frame. Cart A has mass 0.20 kg and moves right at 3.0 m/s. Cart B has mass 0.30 kg and is initially at rest. They stick together. Assume the net external force on the two-cart system is zero during the collision. Find their shared velocity afterward. Right is positive.
  1. Classify the collision
    Because the carts remain together afterward, this is a perfectly inelastic collision. The stated zero net external force allows total momentum conservation for the two-cart system.
    vB,i=0 m/sv_{B,i}=0\,\mathrm{m/s}
  2. Write the momentum relationship
    Use signed velocities. The final mass is the sum of the cart masses because they move together.
    (0.20 kg)(3.0 m/s)+(0.30 kg)(0 m/s)=(0.20+0.30) kg vf(0.20\,\mathrm{kg})(3.0\,\mathrm{m/s})+(0.30\,\mathrm{kg})(0\,\mathrm{m/s})=(0.20+0.30)\,\mathrm{kg}\,v_f
  3. Solve and check
    The initial momentum is positive, so the shared velocity must point right. Dividing total momentum by combined mass gives the final velocity.
    vf=0.60 kg m/s0.50 kg=1.2 m/sv_f=\frac{0.60\,\mathrm{kg\,m/s}}{0.50\,\mathrm{kg}}=1.2\,\mathrm{m/s}
Answer: The carts move together at 1.2 m/s1.2\,\mathrm{m/s} to the right.
Check: The units reduce to metres per second. The final speed is less than cart A's initial speed because the moving cart shares its momentum with the initially resting cart.

Common mistakes and how to avoid them

Treating momentum as a scalar and adding its magnitudes only.
Correction: Momentum is a vector. Use signed velocity components so direction is included.
Assuming every collision conserves kinetic energy.
Correction: Only an elastic collision has unchanged total kinetic energy. Momentum conservation requires a system with zero net external force.
Calling force multiplied by distance impulse.
Correction: Force multiplied by displacement along the force is work. Impulse uses force multiplied by a time interval.
Reporting a negative kinetic energy because velocity is negative.
Correction: Kinetic energy is a scalar and depends on speed squared. A negative velocity indicates direction, not negative kinetic energy.

Lesson summary

Check your understanding

Question 1

A 2.0 kg object moves left at 3.0 m/s. If right is positive, what is its momentum?
  1. +6.0 kg m/s+6.0\,\mathrm{kg\,m/s}
  2. −6.0 kg m/s-6.0\,\mathrm{kg\,m/s}
  3. −1.5 kg m/s-1.5\,\mathrm{kg\,m/s}
  4. +1.5 kg m/s+1.5\,\mathrm{kg\,m/s}
Show answer and explanation
−6.0 kg m/s-6.0\,\mathrm{kg\,m/s}
Momentum equals mass times signed velocity. Leftward velocity is negative, so the momentum is (2.0 kg)(−3.0 m/s)=−6.0 kg m/s(2.0\,\mathrm{kg})(-3.0\,\mathrm{m/s})=-6.0\,\mathrm{kg\,m/s}.

Question 2

A constant net force of 4.0 N acts for 0.50 s. What is the impulse magnitude?
  1. 0.125 N s0.125\,\mathrm{N\,s}
  2. 2.0 N s2.0\,\mathrm{N\,s}
  3. 4.5 N s4.5\,\mathrm{N\,s}
  4. 8.0 N s8.0\,\mathrm{N\,s}
Show answer and explanation
2.0 N s2.0\,\mathrm{N\,s}
Impulse magnitude is force magnitude times time: (4.0 N)(0.50 s)=2.0 N s(4.0\,\mathrm{N})(0.50\,\mathrm{s})=2.0\,\mathrm{N\,s}.

Question 3

Two objects stick together after a collision. Which term describes this collision?
  1. Perfectly inelastic
  2. Elastic
  3. Zero impulse
  4. Zero momentum
Show answer and explanation
Perfectly inelastic
Objects that remain together after a collision undergo a perfectly inelastic collision. This description does not by itself mean that the system's momentum is zero.

Key terms

System
The object or group of objects chosen for study.
Reference frame
The viewpoint used to describe position and motion.
Work
Energy transferred when a force acts through a displacement.
Kinetic energy
Energy an object has because it is moving.
Momentum
A vector quantity equal to an object's mass multiplied by its velocity.
Impulse
The effect of a net force acting over a time interval; it equals the change in momentum.
Collision
An interaction in which objects exert forces on each other over a time interval.
Conservation of momentum
The total momentum of a system remains constant when its net external force is zero.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C2.1. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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