The Spacetime Metric
Level 2 · Secondary physicsGrades 10–12About 8 hours

Mechanics and conservation laws

Turn motion into predictions using forces, momentum, energy, and symmetry.

Build the quantitative backbone needed to evaluate machines, collisions, oscillators, propulsion claims, and every later energy ledger.

Established foundations

Before you begin

  • • Level 1 foundations
  • • Solving one-step algebraic equations
  • • Reading position and velocity graphs

By the end, you can

  • • Use Newton's laws to predict motion.
  • • Apply momentum conservation to interacting systems.
  • • Build complete kinetic and potential energy ledgers.
  • • Distinguish force, impulse, power, and energy.

Interactive model

Explore before calculating

A body accelerating while a field-like wake trails behind it.
Mechanics first describes the measured acceleration and force. Later courses can compare proposed microscopic explanations of inertia.

Live laboratory

Collision conservation bench

Change two carts and the coefficient of restitution. Momentum closes across the chosen system while kinetic energy reveals what the collision stores or disperses.

beforeafter

Final A: 2.00 m/s

Final B: 2.00 m/s

Momentum residual: 0.0e+0 kg·m/s

Kinetic-energy change: -3.00 J

Momentum remains in the two-cart system; lost kinetic energy becomes deformation, sound, heat, or other internal energy.

Level 2 · Secondary physics teaching kit

Record the investigation. Teach the reasoning.

A learner-facing lab record and a course-specific instructor guide turn the live model into a repeatable classroom investigation.

Learner record

Collision momentum and energy audit

Which quantities remain conserved when two carts collide, and what changes when they stick or rebound?

Download learner record

Instructor guide

Teach for evidence, not button pushing

Learners use a declared system boundary to distinguish momentum conservation from kinetic-energy conservation.

Download instructor guide
Open the complete print-friendly teaching kit →

Lesson 1 of 3

Motion, acceleration, and net force

Which interaction changes an object's velocity, and by how much?

Velocity includes direction; acceleration is the rate at which velocity changes. Newton's second law connects that change to the net external force.

A force diagram must include real interactions acting on the chosen object. Motion itself is not a force, and balanced forces can accompany constant nonzero velocity.

velocityaccelerationnet forceinertial mass

Worked example

A 4 kg cart experiences 12 N forward and 4 N backward. Find its acceleration.

  1. 1. Choose forward as positive.
  2. 2. Net force is 12 − 4 = 8 N.
  3. 3. Use a = Fnet/m = 8/4.

The cart accelerates forward at 2 m/s².

Try it

Video motion graph

Materials: A rolling object, phone video, ruler, and equal time marks.

  1. 1. Record motion beside a ruler.
  2. 2. Mark position at equal time intervals.
  3. 3. Plot position versus time.
  4. 4. Identify where slope and acceleration change.

Notice: A steeper position graph means greater speed; changing slope reveals acceleration.

Check your understanding: A puck moves right at constant velocity. What is its net force?

Answer: Zero.

Constant velocity means zero acceleration, so the vector sum of external forces is zero.

Lesson 2 of 3

Momentum, impulse, and system boundaries

What remains unchanged when objects exchange forces internally?

Momentum p = mv is conserved for a system with negligible external impulse. Internal forces change each object's momentum but cancel in the total.

Impulse FΔt equals momentum change. Extending collision time can reduce peak force without changing the required momentum change.

momentumimpulsesystemexternal force

Worked example

A 2 kg cart at 3 m/s sticks to a stationary 1 kg cart.

  1. 1. Initial momentum is 2×3 = 6 kg·m/s.
  2. 2. Combined mass is 3 kg.
  3. 3. Set 3v = 6.

They move together at 2 m/s; kinetic energy is not conserved in the sticking collision.

Try it

Collision system audit

Materials: Two toy carts or coins and a phone camera.

  1. 1. Choose the two-object system.
  2. 2. Record before and after velocities.
  3. 3. Estimate total momentum.
  4. 4. List friction and other external impulses.

Notice: Momentum agreement improves when the system boundary includes both interacting objects and external impulses are small.

Check your understanding: Why does an airbag reduce injury force?

Answer: It spreads the same momentum change over more time.

For a fixed impulse, increasing Δt lowers the average force.

Lesson 3 of 3

Energy conservation and power

How quickly is energy transferred, and does the full ledger close?

Work transfers energy when a force acts through displacement. Kinetic and potential energy track motion and configuration; thermal energy records dispersed microscopic motion.

Power is energy transferred per time. A high-power device may deliver energy quickly without creating more total energy.

workkinetic energypotential energypower

Worked example

A 5 kg load rises 2 m in 4 s. Ignore losses and use g ≈ 9.8 m/s².

  1. 1. Gain in gravitational energy is mgh.
  2. 2. Compute 5×9.8×2 = 98 J.
  3. 3. Power is 98/4.

The average mechanical power is 24.5 W.

Try it

Human power measurement

Materials: Stairs, stopwatch, and a mass estimate.

  1. 1. Measure vertical stair height.
  2. 2. Time a safe ascent.
  3. 3. Compute mgh.
  4. 4. Divide by time and compare repeat trials.

Notice: Faster ascent raises power while the gravitational energy change for the same person and height stays nearly fixed.

Check your understanding: Can a device output more power for a short time than its source supplies at that instant?

Answer: Yes, if stored energy is decreasing.

Peak power can come from a battery, capacitor, flywheel, or other store; the complete energy ledger must include that change.

Formula-to-meaning deck

Read the equation in ordinary language.

F_net = ma

Net external force equals mass times acceleration.

Units: N = kg·m/s²

p = mv; J = Δp

Momentum is mass times velocity; impulse changes momentum.

Units: kg·m/s

K = ½mv²; P = ΔE/Δt

Kinetic energy grows with speed squared; power measures transfer rate.

Units: J; W

Independent practice

Problem set

Work each problem before opening its hint and solution.

  1. 1. A 1200 kg car accelerates at 2.5 m/s². What net force acts?

    Reveal hint

    Use F = ma.

    Reveal solution

    3000 N in the acceleration direction.

  2. 2. A 0.15 kg ball changes velocity from +20 to −10 m/s. Find Δp.

    Reveal hint

    Subtract final minus initial velocity before multiplying by mass.

    Reveal solution

    Δp = 0.15(−10−20) = −4.5 kg·m/s.

  3. 3. A 60 W lamp runs for 5 minutes. How much energy is transferred?

    Reveal hint

    Convert minutes to seconds and use E = Pt.

    Reveal solution

    60×300 = 18,000 J.

Continue into the evidence

Source-linked next reading

Chapter 1: Evidence ladder

Use conservation residuals as a disciplined test of device claims.

Lecture 9: ZPF inertia program

Compare established mechanics with a contested microscopic inertia proposal.

Precision propulsion metrology

Force measurement under vibration, thermal drift, and electromagnetic coupling.