C2.6 Analyse elastic and inelastic collisions in one and two dimensions

Ontario Grade 12 Physics

Energy and Momentum

Momentum and kinetic energy in one and two dimensions

In SPH3U, you learned that velocity has both magnitude and direction, while speed has magnitude only. You also used Newton's laws to connect forces and motion. In this lesson, you will use momentum to analyse collisions. Momentum is a vector: its direction matters. Kinetic energy is a scalar: it has magnitude but no direction. The chosen system and reference frame matter because they determine which objects are included and how their velocities are measured. All examples use ordinary, non-relativistic speeds.

What you will learn

Describe a collision using a defined system, reference frame, and coordinate directions.
Use conservation of momentum to analyse collisions in one dimension and in two dimensions.
Distinguish elastic collisions from inelastic collisions by comparing kinetic energy before and after.
Resolve two-dimensional velocities into components and interpret the final velocity's direction.

1. Define the system and describe the collision

A system is the object or group of objects being studied. For a collision, choose both colliding objects as the system. During a short collision, external forces such as friction may be small compared with the forces the objects exert on each other. If the net external impulse is negligible, the system's total momentum is conserved. Impulse is a force acting over a time interval; here, the key idea is that a negligible external impulse means no important change in the system's total momentum.

A reference frame is the viewpoint used to measure positions and velocities. Use a frame at rest relative to the floor unless a problem specifies another one. In one dimension, choose a positive direction before assigning signs. A velocity opposite to that direction is negative. In two dimensions, choose perpendicular

x

and

y

axes. Each velocity can then be described by its horizontal and vertical components.

An elastic collision conserves both total momentum and total kinetic energy. An inelastic collision conserves total momentum, but the system's total kinetic energy decreases. Some kinetic energy may become sound, thermal energy, or deformation. In a perfectly inelastic collision, the objects stick together and share one final velocity. Do not assume that objects stick together in every inelastic collision.

\vec p=m\vec v

Momentum is a vector; kinetic energy is a scalar.
Use signed velocities in one dimension and components in two dimensions.
The system's momentum is conserved when external impulse is negligible.

2. Apply the collision relationships

Momentum is mass multiplied by velocity. Its SI unit is kilogram metre per second, written kg·m/s. For two objects, add their momentum vectors to get total momentum. Conservation of momentum says that the total before the collision equals the total after it, provided the system has negligible external impulse.

Kinetic energy is the energy of motion. Its SI unit is the joule, J. To test whether a collision is elastic, compare the sum of the objects' kinetic energies before and after. Equal totals indicate an elastic collision in this model. A smaller total after the collision indicates an inelastic collision.

In one dimension, write momentum with signed velocities. In two dimensions, apply momentum conservation separately along the

x

-axis and the

y

-axis. These component equations must both hold. If objects stick together, use their combined mass and one shared final velocity. Use algebra to solve for unknown velocities; do not apply kinetic-energy conservation to an inelastic collision.

\sum \vec p_{\mathrm{before}}=\sum \vec p_{\mathrm{after}}

Momentum conservation applies to both elastic and inelastic collisions.
Kinetic energy is conserved only in an elastic collision.
In two dimensions, conserve momentum independently along each axis.

3. Use components to analyse two-dimensional motion

A vector diagram helps keep directions clear. Draw each velocity arrow from the collision point, then show the chosen positive axes. For calculations, resolve each velocity into perpendicular components. If a velocity of magnitude

v

makes angle

\theta

measured from the positive

x

-axis, its components are

v_x=v\cos\theta

and

v_y=v\sin\theta

. The signs depend on the direction of the arrow.

After finding the final momentum components, divide by the relevant mass to obtain velocity components. The speed follows from the two components, and the direction can be found with an inverse tangent. State the quadrant or describe the direction in words so that the angle is not ambiguous.

A final check should include units and physical meaning. Momentum components must have units of kg·m/s; velocity components must have units of m/s. Check whether each sign agrees with the diagram. For an inelastic collision, confirm that the final kinetic energy is not greater than the initial kinetic energy under the stated model.

v_x=v\cos\theta,\qquad v_y=v\sin\theta

Angles must be tied to a stated axis.
A negative component means motion opposite to that axis's positive direction.
Use the final component signs to describe the direction.

Worked example

1. Elastic collision in one dimension

A 2.0 kg cart moving at +3.0 m/s collides elastically with a 1.0 kg cart at rest. Find both carts' final velocities. The system is the two carts, the frame is the floor, and right is positive.

Set up momentum
The carts form the system. Their total momentum is conserved. The initial momentum is $2.0\,\mathrm{kg}(+3.0\,\mathrm{m/s})+1.0\,\mathrm{kg}(0\,\mathrm{m/s})$ .
$p_{\mathrm{before}}=+6.0\,\mathrm{kg\cdot m/s}$
Use elastic-collision relationships
Because the collision is elastic, kinetic energy is conserved as well as momentum. Solving the two conservation equations gives a final velocity of +1.0 m/s for the 2.0 kg cart and +4.0 m/s for the 1.0 kg cart.
$v_{1f}=+1.0\,\mathrm{m/s},\qquad v_{2f}=+4.0\,\mathrm{m/s}$

Answer: The 2.0 kg cart moves right at 1.0 m/s, and the 1.0 kg cart moves right at 4.0 m/s.

Check: Final momentum is

2.0(1.0)+1.0(4.0)=6.0\,\mathrm{kg\cdot m/s}

. Initial kinetic energy is 9.0 J, and final kinetic energy is

1.0+8.0=9.0\,\mathrm{J}

. Both conservation checks pass.

Worked example

2. Perfectly inelastic collision in two dimensions

A 0.20 kg puck moves east at 3.0 m/s. It collides with a 0.10 kg puck moving north at 4.0 m/s. They stick together. Find their final velocity. The two-puck system is viewed from the floor; east is +x and north is +y.

Find initial momentum components
The first puck contributes momentum only along +x. The second contributes only along +y. Add the components separately.
$p_x=(0.20)(3.0)=0.60\,\mathrm{kg\cdot m/s},\qquad p_y=(0.10)(4.0)=0.40\,\mathrm{kg\cdot m/s}$
Find the shared final velocity
The pucks stick, so the final mass is 0.30 kg and both have the same velocity. Divide each conserved momentum component by that mass.
$v_x=2.0\,\mathrm{m/s},\qquad v_y=1.3\,\mathrm{m/s}$
Describe the velocity
Use the components to find speed and the angle measured north of east. The positive signs place the motion in the north-east direction.
$v=2.4\,\mathrm{m/s},\qquad \theta=34^\circ$

Answer: The joined pucks move at 2.4 m/s, 34° north of east.

Check: Initial kinetic energy is 1.7 J. Final kinetic energy is approximately 0.87 J, so kinetic energy decreased as expected for a perfectly inelastic collision. The component units reduce to m/s after division by mass.

Worked example

3. Inelastic collision in one dimension without sticking

A 0.50 kg cart moves right at 4.0 m/s and collides inelastically with a 0.30 kg cart moving left at 1.0 m/s. Afterward, the first cart moves right at 1.0 m/s. Find the second cart's final velocity and determine the change in kinetic energy. The system is both carts, the frame is the floor, and right is positive.

Find initial momentum
The second cart's velocity is negative because it moves left. Total momentum is conserved for the two-cart system.
$p_{\mathrm{before}}=(0.50)(+4.0)+(0.30)(-1.0)=+1.7\,\mathrm{kg\cdot m/s}$
Solve for the unknown velocity
Set final momentum equal to initial momentum. Substitute the known final velocity of the first cart and solve for the second cart's velocity.
$1.7=(0.50)(1.0)+(0.30)v_{2f},\qquad v_{2f}=+4.0\,\mathrm{m/s}$
Compare kinetic energies
Kinetic energy uses speed squared, so the negative initial velocity does not make the initial energy negative. Subtract the initial total from the final total to find the change.
$\Delta E_k=2.65\,\mathrm{J}-4.15\,\mathrm{J}=-1.50\,\mathrm{J}$

Answer: The second cart moves right at 4.0 m/s. The system loses 1.50 J of kinetic energy, so the collision is inelastic.

Check: Final momentum is

(0.50)(1.0)+(0.30)(4.0)=1.7\,\mathrm{kg\cdot m/s}

, matching the initial momentum. The energy change is negative, as required for this inelastic collision.

Common mistakes and how to avoid them

Treating momentum as a positive quantity regardless of direction.

Correction: Momentum is a vector. In one dimension, use a positive or negative sign based on the chosen axis.

Assuming kinetic energy is conserved in every collision.

Correction: Kinetic energy is conserved in elastic collisions. In inelastic collisions, total kinetic energy decreases.

Using one momentum equation for a two-dimensional collision.

Correction: Write one conservation equation for the x-components and another for the y-components.

Calling every inelastic collision perfectly inelastic.

Correction: Only a perfectly inelastic collision has objects that stick together and share a final velocity.

Giving a two-dimensional angle without saying what it is measured from.

Correction: Name the axis and direction, such as 34° north of east.

Lesson summary

Define the system, reference frame, and positive axes before assigning velocities.
Conserve total momentum when external impulse is negligible.
Use kinetic-energy conservation only for elastic collisions.
For two-dimensional collisions, conserve momentum separately along perpendicular axes.
Check units, signs, directions, and whether the energy result matches the collision type.

Check your understanding

Question 1

A 1.0 kg object moves at +2.0 m/s and collides with a 1.0 kg object at rest. They stick together. What is their final velocity?

+1.0 m/s
+2.0 m/s
−1.0 m/s
0 m/s

Show answer and explanation

+1.0 m/s

Initial momentum is +2.0 kg·m/s. Divide by the combined mass of 2.0 kg to get +1.0 m/s.

Question 2

Which quantity is conserved in both elastic and inelastic collisions when external impulse is negligible?

Each object's kinetic energy
The system's total momentum
Each object's velocity
The system's total kinetic energy

Show answer and explanation

The system's total momentum

The system's total momentum is conserved. Total kinetic energy is conserved only for an elastic collision.

Question 3

A final momentum component is negative on the y-axis. What does this indicate?

The object has no vertical motion.
The object moves in the positive y-direction.
The object moves opposite to the positive y-direction.
The object's kinetic energy is negative.

Show answer and explanation

The object moves opposite to the positive y-direction.

A negative component points opposite to the chosen positive direction on that axis.

Key terms

System: The object or group of objects selected for analysis.
Reference frame: The viewpoint used to measure position and velocity.
Momentum: A vector quantity equal to an object's mass multiplied by its velocity.
Elastic collision: A collision in which total momentum and total kinetic energy are conserved.
Inelastic collision: A collision in which momentum is conserved but total kinetic energy decreases.
Perfectly inelastic collision: An inelastic collision in which the objects stick together after impact.
Component: One part of a vector along a chosen axis.

Continue through SPH4U

View the complete SPH4U Ontario Grade 12 Physics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C2.6. 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.

Official curriculum reference

Report a correction or ask a question

C2.6 · Analyse elastic and inelastic collisions in one and two dimensions

Ontario Grade 12 Physics

Momentum and kinetic energy in one and two dimensions

What you will learn

1. Define the system and describe the collision

2. Apply the collision relationships

3. Use components to analyse two-dimensional motion

1. Elastic collision in one dimension

2. Perfectly inelastic collision in two dimensions

3. Inelastic collision in one dimension without sticking

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH4U

About this lesson and its review