C3.4 Explain implications of energy and momentum conservation

Ontario Grade 12 Physics

Energy and Momentum

Using system boundaries to predict motion and track energy changes

Conservation laws help predict what can happen during an interaction. They do not say that each object keeps the same energy or momentum. Instead, they describe what remains constant for a chosen system under the right conditions. A system is the object or group of objects being studied. A reference frame is the viewpoint used to measure position and velocity. We will use a ground-based frame unless stated otherwise. Choose a positive direction before using signed velocities. Energy is a scalar: it has magnitude but no direction. Momentum is a vector: it has magnitude and direction, so signs or components matter.

What you will learn

Explain what energy and momentum conservation mean for a defined system.
Use momentum conservation to predict motion when a system has negligible net external force.
Track changes in kinetic energy and identify where energy may be transferred or transformed.
Check calculations using units, direction, and physical reasonableness.

1. The SPH3U bridge: energy, momentum, and systems

A system is the object or group of objects whose changes we track. Its boundary separates what is inside from what is outside. For example, a collision system might include both carts. The forces the carts exert on each other are then internal forces. A force from something outside the system is an external force.

Kinetic energy is energy associated with motion. Mass is measured in kilograms, speed in metres per second, and kinetic energy in joules. Momentum equals mass multiplied by velocity. Since velocity includes direction, momentum does too. Its SI unit is the kilogram metre per second.

A system is isolated from external forces for an interaction when the net external force is zero or small enough to ignore. In that case, total momentum stays constant. For energy, track the full system and all relevant energy forms. Energy can move between objects or change form. Energy crossing the system boundary can change the system's energy, even though energy is conserved overall.

\vec p=m\vec v

Choose and name the system, reference frame, and positive direction before calculating.
Energy is scalar; momentum is vector.
Momentum conservation requires negligible net external force on the chosen system during the interaction.

2. What the conservation rules imply

Momentum conservation means the system's total momentum before an interaction equals its total momentum after, provided the system has negligible net external force during the interaction. This rule applies to the vector total. In one dimension, a chosen positive direction is represented by a positive sign and the opposite direction by a negative sign.

Energy conservation means the total energy of a suitably defined system is accounted for before and after a change. Energy may be transferred between objects or transformed into other forms. In a collision, some initial kinetic energy can become sound, thermal energy, or energy associated with deformation. Kinetic energy alone is therefore not always conserved, even when momentum is conserved.

Together, the rules limit the possible outcomes. Momentum can predict a shared velocity after objects stick together, or the recoil of one object when another moves away. Energy accounting can show whether motion energy has changed and that the change must be accounted for. Do not assume that a decrease in kinetic energy means energy has disappeared.

\vec p_{\text{before}}=\vec p_{\text{after}}

For a system with negligible net external force, total momentum before and after is equal.
Total energy is conserved when all relevant transfers and forms are included in the accounting.
Kinetic energy can change in a collision; momentum conservation does not require it to stay constant.

3. Applying the rules and checking the result

For a one-dimensional interaction, first state the positive direction. List each object's mass and signed velocity before and after. Then use the total momentum relationship. Keep units through the substitution. A negative velocity means motion opposite to the chosen positive direction, not a negative speed.

For energy, list the forms relevant to the system before and after. A useful comparison in a collision is the total kinetic energy before and after. If those values differ, energy has changed form or crossed the system boundary. State which forms are known. Do not claim an exact destination when the information does not specify it.

A final check should ask whether the units match the quantity, whether the direction agrees with the sign, and whether the result is plausible. Conservation rules constrain outcomes, but they do not provide missing details about every energy form.

\sum m_i v_{i,\text{before}}=\sum m_i v_{i,\text{after}}

Use signed velocity for one-dimensional momentum calculations.
Compare kinetic energy separately from total energy.
Check units, direction, and whether the answer fits the physical situation.

4. Reading the implications

A conservation calculation models a defined situation. It is most useful when the system boundary and assumptions are clear. If an external force matters, the momentum of the chosen system may change. If an energy transfer or form is omitted, the energy account will appear incomplete.

The same event can be described using both rules. Momentum describes the direction and amount of motion carried by the system. Energy accounting tracks the amount of energy and its forms. Neither rule says that each object's individual momentum or kinetic energy must remain unchanged. The rules apply to the total for the chosen system.

The result depends on a clearly chosen system and suitable assumptions.
Conservation applies to totals, not necessarily to each object separately.
Momentum and energy provide related but different information.

Worked example

Two carts stick together

A

2.0\,\mathrm{kg}

cart moves right at

3.0\,\mathrm{m/s}

. A

1.0\,\mathrm{kg}

cart moves left at

1.0\,\mathrm{m/s}

. They collide and stick together. Find their final velocity and compare total kinetic energy before and after. Treat the carts as the system and ignore net external force during the collision.

Set the direction and identify the system
Use the ground frame and define right as positive. The system contains both carts. Their interaction forces are internal, and the problem says to neglect net external force. The unknown is the shared final velocity.
$v_{1,i}=+3.0\,\mathrm{m/s},\quad v_{2,i}=-1.0\,\mathrm{m/s}$
Apply momentum conservation
The signed initial momentum is the sum of the two cart momenta. After they stick, their combined mass moves with one velocity. Equating the totals gives the final velocity.
$(2.0\,\mathrm{kg})(3.0\,\mathrm{m/s})+(1.0\,\mathrm{kg})(-1.0\,\mathrm{m/s})=(3.0\,\mathrm{kg})v_f$
Calculate and interpret the velocity
The total initial momentum is $5.0\,\mathrm{kg\,m/s}$ . Dividing by the combined mass gives a positive velocity, so the joined carts move right.
$v_f=\frac{5.0\,\mathrm{kg\,m/s}}{3.0\,\mathrm{kg}}=1.7\,\mathrm{m/s}$
Compare kinetic energies
Kinetic energy is scalar, so add the carts' positive kinetic energies. Use the unrounded shared velocity for the final energy calculation. The difference is energy transformed into other forms, such as sound, thermal energy, or deformation; it has not vanished.
$K_i=\tfrac12(2.0\,\mathrm{kg})(3.0\,\mathrm{m/s})^2+\tfrac12(1.0\,\mathrm{kg})(1.0\,\mathrm{m/s})^2=9.5\,\mathrm{J},\quad K_f=\tfrac12(3.0\,\mathrm{kg})(1.666\ldots\,\mathrm{m/s})^2=4.166\ldots\,\mathrm{J}\approx4.2\,\mathrm{J}$

Answer: The carts move together at

1.7\,\mathrm{m/s}

to the right. Their kinetic energy decreases from

9.5\,\mathrm{J}

to about

4.2\,\mathrm{J}

. The decrease is about

5.3\,\mathrm{J}

.

Check: The initial momentum is

5.0\,\mathrm{kg\,m/s}

, and the final momentum using the unrounded velocity is also

5.0\,\mathrm{kg\,m/s}

. The momentum unit is correct. The final velocity is positive, so it is rightward. The kinetic-energy decrease is not a loss of total energy; it must be accounted for in other forms or transfers.

Worked example

A falling object changes energy form

A

0.80\,\mathrm{kg}

object starts from rest and falls

5.0\,\mathrm{m}

. Find its speed just before reaching the ground if air resistance is negligible. Use

g=9.8\,\mathrm{m/s^2}

.

Define the system and known values
Use the object and Earth as the system in a ground-based frame. Take downward as positive for direction. The object starts at rest, and its gravitational energy becomes kinetic energy when air resistance is negligible. The unknown is its final speed.
$m=0.80\,\mathrm{kg},\quad h=5.0\,\mathrm{m},\quad v_i=0,\quad g=9.8\,\mathrm{m/s^2}$
Relate the energy forms
With negligible air resistance, the decrease in gravitational potential energy equals the increase in kinetic energy. The mass appears on both sides, so it cancels when solving for speed.
$mgh=\tfrac12mv_f^2$
Solve for speed
Substitute the height and gravitational field strength, then take the positive square root because speed is a non-negative scalar.
$v_f=\sqrt{2gh}=\sqrt{2(9.8\,\mathrm{m/s^2})(5.0\,\mathrm{m})}=9.9\,\mathrm{m/s}$

Answer: The object's speed just before reaching the ground is

9.9\,\mathrm{m/s}

.

Check: The quantity inside the square root has units of

\mathrm{m^2/s^2}

, so the result is in

\mathrm{m/s}

. The velocity is downward under the chosen convention; the requested speed is positive. A fall of several metres producing a speed of about

10\,\mathrm{m/s}

is reasonable under the stated assumption.

Worked example

Recoil after an object is thrown

A

50\,\mathrm{kg}

skater and a

2.0\,\mathrm{kg}

object are initially at rest on a low-friction surface. The skater throws the object so it moves right at

5.0\,\mathrm{m/s}

in the ground frame. Find the skater's velocity immediately after the throw. Treat the skater and object as the system and neglect external horizontal force.

Choose the frame and direction
Use the ground frame and define right as positive. The skater and object together begin at rest, so the system's initial horizontal momentum is zero. The unknown is the skater's signed final velocity.
$p_i=0,\quad v_{o,f}=+5.0\,\mathrm{m/s}$
Use total momentum conservation
With negligible net external horizontal force, the final momenta must add to zero. The skater must move opposite to the thrown object so the total remains unchanged.
$(50\,\mathrm{kg})v_{s,f}+(2.0\,\mathrm{kg})(5.0\,\mathrm{m/s})=0$
Solve for the skater's velocity
The object has $10\,\mathrm{kg\,m/s}$ of rightward momentum. The skater must have the same magnitude of momentum to the left. Dividing by the skater's mass gives the velocity.
$v_{s,f}=-\frac{(2.0\,\mathrm{kg})(5.0\,\mathrm{m/s})}{50\,\mathrm{kg}}=-0.20\,\mathrm{m/s}$

Answer: The skater moves at

0.20\,\mathrm{m/s}

to the left.

Check: The negative sign means left under the chosen convention. The skater's momentum is

-10\,\mathrm{kg\,m/s}

, which balances the object's

+10\,\mathrm{kg\,m/s}

. The units reduce to velocity units, and the skater's speed is lower than the object's, as expected for the much larger mass.

Common mistakes and how to avoid them

Saying that kinetic energy is always conserved in a collision.

Correction: Momentum may be conserved while kinetic energy changes. Account for energy transformed into other forms or transferred across the system boundary.

Ignoring direction when adding momentum.

Correction: Choose a positive direction and use signed velocities. Momentum is a vector, so opposite directions contribute opposite signs.

Treating energy or momentum of each object as individually constant.

Correction: Conservation rules describe the total for the chosen system. Objects can exchange momentum and energy with one another.

Claiming energy disappeared when the system's kinetic energy decreases.

Correction: A decrease in kinetic energy means energy changed form or crossed the system boundary. Identify what information is known before naming its destination.

Lesson summary

Define the system, reference frame, and positive direction before applying a conservation rule.
Total momentum remains constant when net external force on the system is negligible.
Energy is conserved when the full energy account includes relevant forms and transfers.
Kinetic energy can decrease during an interaction even when momentum is conserved.
Check units, signs, direction, and physical reasonableness.

Check your understanding

Question 1

Two objects form a system with negligible net external force during a collision. Which quantity must have the same total value before and after?

The momentum of each object
The total momentum of the system
The kinetic energy of each object
The speed of each object

Show answer and explanation

The total momentum of the system

The system's total momentum is conserved under the stated condition. Individual objects can change momentum during the collision.

Question 2

A moving object sticks to a stationary object. The system's kinetic energy after the collision is lower. What does this show?

Energy has been destroyed.
Momentum conservation has failed.
Some kinetic energy has changed form or been transferred.
The objects must have gained speed.

Show answer and explanation

Some kinetic energy has changed form or been transferred.

Kinetic energy is not necessarily conserved in a collision. The energy difference must be accounted for through other forms or transfers.

Question 3

A system starts with zero total momentum. One object moves right after an interaction. If external forces are negligible, what must be true of the other object's momentum?

It must be zero.
It must point right and have equal magnitude.
It must point left and have equal magnitude.
It must have the same speed.

Show answer and explanation

It must point left and have equal magnitude.

The final vector total must remain zero, so the other object's momentum must balance the rightward momentum with equal magnitude to the left.

Key terms

System: The object or group of objects chosen for tracking physical changes.
Reference frame: The viewpoint used to describe positions and motion.
Momentum: A vector quantity equal to an object's mass multiplied by its velocity.
Kinetic energy: Energy associated with an object's motion.
External force: A force on the system from something outside its boundary.
Conservation: A quantity's total value remains constant for a system when the required conditions are met.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C3.4. 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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C3.4 · Explain implications of energy and momentum conservation

Ontario Grade 12 Physics

Using system boundaries to predict motion and track energy changes

What you will learn

1. The SPH3U bridge: energy, momentum, and systems

2. What the conservation rules imply

3. Applying the rules and checking the result

4. Reading the implications

Two carts stick together

A falling object changes energy form

Recoil after an object is thrown

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