Difficulty: Introductory to Intermediate | Prerequisites: Kinematics, Newton's laws, and basic algebra.
This unit ties together everything from the first half of the course. Work connects force to energy, and the work-energy theorem provides an alternative to F = ma for solving problems. Conservation of energy lets you bypass complicated force analyses when only the start and end states matter. Momentum and collisions bring in a new conserved quantity, and impulse links force to changes in momentum over time. If Newton's second law made sense, this unit builds naturally on top of it.
Work equals force times displacement (when parallel). Kinetic energy depends on speed squared, so doubling speed quadruples KE. Conservation of energy says total mechanical energy stays constant when only conservative forces act. Momentum is always conserved in collisions; kinetic energy is conserved only in elastic collisions.
Work (W)
The energy transferred to or from an object by a force acting over a displacement: W = F d cos θ, where θ is the angle between the force and the displacement. In simple terms, work is done when a force moves something.
Kinetic energy (KE)
The energy an object has because of its motion: KE = ½ mv². Think of it as the energy it took to get the object moving to its current speed.
Potential energy (PE)
Stored energy due to position or configuration. Gravitational PE = mgh, where h is the height above a reference point.
Conservation of energy
In a closed system with no non-conservative forces (like friction), the total mechanical energy (KE + PE) remains constant. Energy is neither created nor destroyed, only converted between forms.
Momentum (p)
The product of mass and velocity: p = mv. Momentum is a vector. In simple terms, a heavy object moving slowly can have the same momentum as a light object moving quickly.
Impulse (J)
The change in momentum of an object, equal to the average force times the time interval: J = F_avg × Δt = Δp. Think of impulse as the "push" accumulated over time.
Elastic collision
A collision in which both momentum and kinetic energy are conserved. The objects bounce off each other with no energy lost to deformation, heat, or sound.
Inelastic collision
A collision in which momentum is conserved but kinetic energy is not. Some kinetic energy is converted to other forms (heat, sound, deformation). A perfectly inelastic collision is one where the objects stick together.
Average force
The constant force that would deliver the same impulse over the same time interval as the varying force. For a triangular force-time pulse, the average force equals half the peak force.
Work = F d cos θ, where F is the force magnitude, d is the displacement, and θ is the angle between them.
When force and displacement are in the same direction (θ = 0°), W = Fd.
Example: pushing a 5.0 kg crate 4.0 m with a horizontal force of 20 N on a frictionless floor gives W = 20 × 4.0 = 80 J.
When force is perpendicular to displacement (θ = 90°), W = 0. The normal force and gravity on a horizontally sliding object do no work.
Negative work means the force opposes the motion (θ > 90°), such as friction slowing an object.
KE = ½ mv².
Because velocity is squared, doubling the speed quadruples the kinetic energy.
If v becomes 2v: KE_new = ½ m(2v)² = ½ m(4v²) = 4 × (½ mv²) = 4 × KE_original.
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy: W_net = ΔKE.
When only conservative forces (gravity, springs) do work, total mechanical energy is conserved: KE_i + PE_i = KE_f + PE_f.
This is powerful because you only need the initial and final states, not the details of the path.
Example: a child on a swing (30 kg) starts from rest at height 2.0 m and swings down to height 0.5 m. Using conservation of energy:
mgh_i = ½ mv² + mgh_f.
gh_i = ½ v² + gh_f (mass cancels).
v = √(2g(h_i - h_f)) = √(2 × 9.8 × 1.5) = √(29.4) ≈ 5.42 m/s.
Notice that mass cancels in many conservation-of-energy problems. The speed at the bottom depends only on the height difference, not on the child's weight.
Momentum: p = mv (a vector).
Impulse: J = F_avg × Δt = Δp.
Conservation of momentum: in any collision (or explosion), total momentum before = total momentum after, provided no external net force acts.
Perfectly inelastic collision (objects stick together):
m₁v₁ + m₂v₂ = (m₁ + m₂)v_f.
Example: a 2.0 kg block at 4.0 m/s hits and sticks to a stationary 2.0 kg block. v_f = (2.0 × 4.0) / (2.0 + 2.0) = 8.0 / 4.0 = 2.0 m/s.
Elastic collision: both momentum and kinetic energy are conserved. Objects bounce apart.
Inelastic collision: momentum is conserved, but kinetic energy is not. Some KE is lost to heat, sound, or deformation.
The primary difference between elastic and inelastic collisions is whether kinetic energy is conserved. Momentum is conserved in both.
Average force from a force-time graph:
The impulse equals the area under the F vs t curve.
For a triangular pulse from t = 0 to t = 0.10 s with peak F = 1800 N: area = ½ × base × height = ½ × 0.10 × 1800 = 90 N·s.
Average force = impulse / Δt = 90 / 0.10 = 900 N.
Strobe (motion) diagrams show an object's position at equal time intervals.
Increasing spacing between dots = speeding up.
Decreasing spacing = slowing down.
The object stops where dots converge.
For a bungee jumper:
Instants 1 to 4: falling downward, spacing increases = accelerating downward (gravity dominates).
Instants 4 to 6: falling downward, spacing decreases = decelerating (bungee cord pulls upward, opposing motion) until the jumper stops at instant 6.
Instants 6 to 10: moving upward, spacing increases then starts to decrease.
Instants 10 to 12: moving upward, spacing decreases = slowing down until the jumper stops at instant 12.
Average acceleration between instants 9 and 12: the jumper is moving upward and slowing down (spacing decreases). Since the velocity is upward but decreasing in magnitude, the acceleration must be directed downward.
At instant 6 (the lowest point), the jumper has stopped and is about to reverse direction upward. The acceleration is directed upward (the jumper begins moving upward immediately after). For the acceleration to be upward, the net force must be upward, which means the bungee cord's upward force exceeds the jumper's weight. The cord force is greater than the gravitational force at this point.
Work: W = Fd cos θ
Kinetic energy: KE = ½ mv²
Gravitational PE: PE = mgh
Conservation of energy: KE_i + PE_i = KE_f + PE_f
Momentum: p = mv
Impulse: J = F_avg Δt = Δp
Impulse from graph: J = area under F vs t curve
Perfectly inelastic collision: m₁v₁ + m₂v₂ = (m₁ + m₂)v_f
Conservation of energy is how roller-coaster designers calculate the speed at the bottom of a hill without knowing anything about the forces along the track. Airbags work by increasing the collision time Δt, which reduces the average force F_avg for the same impulse (change in momentum). Crumple zones in cars serve the same purpose.
Students often think doubling speed doubles kinetic energy. Because KE depends on v squared, doubling speed quadruples KE.
Students confuse conservation of momentum with conservation of kinetic energy. Momentum is conserved in all collisions. Kinetic energy is conserved only in elastic collisions.
Students assume that a heavier object always has more kinetic energy. A lighter object moving faster can have more KE than a heavy slow one.
Students sometimes think that at the lowest point of a bungee jump, the forces are balanced (net force = 0). They are not; the bungee force exceeds weight, producing a net upward force and upward acceleration.
⚠️ The v-squared relationship in KE is tested frequently. "If speed doubles, what happens to KE?" is almost guaranteed.
⚠️ Conservation of energy problems (like the swing) are common. Remember that mass often cancels.
⚠️ Perfectly inelastic collision problems (objects stick together) appear regularly. Use m₁v₁ = (m₁ + m₂)v_f when one object is initially stationary.
⚠️ Impulse from a force-time graph: know how to find the area (triangle = ½ base × height) and divide by Δt for average force.
⚠️ Strobe diagram analysis requires you to connect dot spacing to velocity changes and then to acceleration direction. Practise this reasoning.
Fill in the blank: if speed triples, kinetic energy increases by a factor of ______. (9)
True or false: momentum is conserved in an inelastic collision. (True)
Fill in the blank: the area under a force-time graph equals the ______. (impulse)
True or false: in a perfectly inelastic collision, the objects bounce apart. (False, they stick together.)
Fill in the blank: at the lowest point of a bungee jump, the bungee force is ______ than the weight. (greater)
Q: A 5.0 kg crate is pushed 4.0 m across a frictionless floor by a constant horizontal force of 20 N. What is the work done by the pushing force?
A: W = Fd cos θ = 20 × 4.0 × cos 0° = 80 J. Force and displacement are in the same direction, so cos 0° = 1.
Q: If the speed of a particle is doubled, what happens to its kinetic energy?
A: It quadruples. KE = ½ mv², so replacing v with 2v gives ½ m(2v)² = 4 × ½ mv².
Q: A 2.0 kg block moving at 4.0 m/s collides with and sticks to a stationary 2.0 kg block. What is the final velocity?
A: Using conservation of momentum: (2.0)(4.0) + (2.0)(0) = (2.0 + 2.0)v_f. So v_f = 8.0 / 4.0 = 2.0 m/s.
Q: A child on a swing (30 kg) starts from rest at 2.0 m above the ground and swings to a lowest point at 0.5 m. What is their speed at the bottom?
A: v = √(2g Δh) = √(2 × 9.8 × 1.5) = √29.4 ≈ 5.42 m/s. Mass cancels out of the conservation-of-energy equation.
Q: A 4.5 kg head hits an airbag. The force-time graph is a triangular pulse from t = 0 to t = 0.10 s, peaking at 1800 N at t = 0.05 s. What is the average force?
A: Impulse = area of triangle = ½ × 0.10 × 1800 = 90 N·s. Average force = 90 / 0.10 = 900 N.
Q: Define an elastic collision and an inelastic collision. What is the primary difference?
A: In an elastic collision, both momentum and kinetic energy are conserved; the objects bounce apart without any loss of KE. In an inelastic collision, momentum is conserved but kinetic energy is not; some KE is converted into heat, sound, or deformation. The primary difference is whether kinetic energy is conserved.
Q: A bungee jumper is at the lowest point (instant 6 in the strobe diagram). Is the bungee cord force greater than, less than, or equal to the jumper's weight?
A: Greater than. At the lowest point the jumper has stopped and is about to accelerate upward, so the net force must be directed upward. This means the upward bungee force exceeds the downward gravitational force.
Q: Between instants 9 and 12 in the bungee strobe diagram, the jumper moves upward with decreasing spacing. What is the direction of the average acceleration?
A: Downward. The jumper is moving upward but slowing down (decreasing dot spacing means decreasing speed). A decrease in upward velocity means the acceleration opposes the motion, so it points downward.
Work and energy provide an alternative problem-solving framework to Newton's second law. Many problems that are difficult with forces alone become straightforward with energy conservation, particularly those involving curved paths or variable forces. Momentum conservation connects to Newton's third law: the equal-and-opposite force pairs between colliding objects ensure that the total momentum of the system cannot change. Impulse ties back to force-time analysis and is the bridge between Newton's second law (F = ma = m Δv/Δt) and momentum (Δp = FΔt). These ideas extend later into rotational dynamics, where torque, angular momentum, and rotational kinetic energy follow analogous patterns.
work, energy, kinetic energy, potential energy, gravitational potential energy, conservation of energy, work-energy theorem, momentum, impulse, elastic collision, inelastic collision, perfectly inelastic collision, average force, force-time graph, impulse-momentum theorem, strobe diagram, motion diagram, bungee jumper physics, collision types, PHYS 114, general physics, introductory mechanics, University of Washington