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Energy & Momentum — Visual Lab
Module 3. Two carts on a frictionless track, one collision. Set the masses and the velocities, choose whether they bounce or stick, and run it — then read what survives the crash. Momentum always balances; kinetic energy only survives when the collision is elastic.
A moving object carries two stockpiles at once: momentum \(p = mv\) and kinetic energy \(KE = \tfrac12 mv^2\). When two objects collide, momentum is always conserved — the total before equals the total after, no exceptions. Kinetic energy is pickier: an elastic collision keeps all of it, while a perfectly inelastic one (the carts couple) sheds the maximum it can to heat, sound, and deformation. Drive the bench below and watch both ledgers update in real numbers.
1-D Collision Bench
Set each cart's mass and velocity (right is \(+\), left is \(-\)), pick elastic or perfectly inelastic, and press Run. The before/after panel computes the total momentum \(p = m_1u_1 + m_2u_2\) and the total kinetic energy \(KE = \tfrac12 m_1u_1^2 + \tfrac12 m_2u_2^2\) on both sides of the collision.
What you're seeing
- Each cart carries an arrow. Its direction is the sign of the velocity (right \(=+\), left \(=-\)) and its length grows with speed. The number on it is that cart's velocity in m/s.
- Total momentum is the same line, before and after. The highlighted \(p\) row in the after-panel matches the before-panel — that's the conservation law you can see.
- Kinetic energy may shrink. In elastic mode the \(KE\) rows match; in inelastic mode the after-panel shows less energy, and the conservation check names how much was lost.
- Inelastic carts couple. Switch off "elastic" and the carts leave the collision joined by a bar, sharing one velocity \(v = \dfrac{m_1u_1 + m_2u_2}{m_1+m_2}\).
- The track is frictionless. Between collisions the carts glide at constant velocity — no slowing — so every change you see comes from the collision itself.
Try this
- Equal masses, head-on, elastic. Set \(m_1 = m_2\), \(u_1 = +4\), \(u_2 = -4\). The carts swap velocities exactly — the signature of an equal-mass elastic hit.
- Heavy hits light. Make \(m_1 = 6\), \(m_2 = 0.5\), \(u_2 = 0\). The light cart rockets off at nearly twice the heavy cart's speed — check the after-panel.
- Flip to inelastic. Keep the same setup and uncheck "elastic." Watch the \(KE\) row drop and the conservation check report the percentage lost — while \(p\) stays put.
- Same direction, catch-up. Try \(u_1 = +6\), \(u_2 = +2\) (cart 1 behind, faster). Compare the elastic bounce to the inelastic coupling — same momentum, very different energy.
Worked Examples
Two moves this module asks for most: conserving momentum through a perfectly inelastic collision, and checking how much kinetic energy that collision costs.
Example 1 — A perfectly inelastic collision (carts stick)
A \(2\text{ kg}\) cart moving at \(+6\text{ m/s}\) strikes and couples to a \(3\text{ kg}\) cart at rest. Find the velocity of the joined carts.
- Write conservation of momentum. Total before \(=\) total after: \(m_1u_1 + m_2u_2 = (m_1+m_2)v\).
- Substitute. \(2(6) + 3(0) = (2+3)\,v\), so \(12 = 5v\).
- Solve for \(v\). Divide both sides by \(5\): \(v = \dfrac{12}{5} = 2.4\text{ m/s}\).
- Sanity-check the sign. Positive, so the joined carts move to the right — the same direction the moving cart was going. Good.
Example 2 — How much kinetic energy did that cost?
Using the carts from Example 1, find the kinetic energy before and after, then the amount lost.
- \(KE\) before. Only cart 1 moves: \(KE_i = \tfrac12 m_1 u_1^2 = \tfrac12 (2)(6)^2\). Square first: \(6^2 = 36\); then \(\tfrac12(2)(36) = 36\text{ J}\).
- \(KE\) after. Both carts move together at \(2.4\text{ m/s}\): \(KE_f = \tfrac12 (m_1+m_2) v^2 = \tfrac12 (5)(2.4)^2\). Square: \(2.4^2 = 5.76\); then \(\tfrac12(5)(5.76) = 14.4\text{ J}\).
- Energy lost. \(\Delta KE = 36 - 14.4 = 21.6\text{ J}\) — gone to heat, sound, and bending metal.
- Confirm momentum didn't move. Before: \(2(6) = 12\). After: \(5(2.4) = 12\). Equal \(\checkmark\) — energy can vanish while momentum cannot.
Key Vocabulary
The precise words a physicist uses to describe what the bench is doing.
\(p = mv\), a vector. The "amount of motion" an object carries; the total for a closed system never changes (kg·m/s).
\(KE = \tfrac12 mv^2\), a scalar. The energy of motion. Depends on the square of speed, so doubling \(v\) quadruples \(KE\) (joules).
A collision in which total kinetic energy is conserved as well as momentum. The objects bounce apart with no energy lost.
A collision that conserves momentum but loses kinetic energy. In the perfectly inelastic case the objects stick together.
\(J = F\,\Delta t\), the change in momentum delivered by a force over time: \(J = \Delta p\). A bigger force or a longer push changes motion more.
Work \(W = Fd\cos\theta\) is energy transferred by a force; power \(P = \dfrac{W}{t}\) is how fast that energy is delivered (watts).
Standards & honesty
This lab is Physics · enrichment — not a TEKS-graded course. The conservation of momentum and energy belongs to high-school physics (TEKS 19 TAC §112.39, e.g. P.6 momentum and Newton's laws, P.7 work and energy), a separate subject from the Bluebonnet mathematics sequence these studio courses follow. Treat these pages as an applied playground for algebra: a place to square numbers, solve a formula for a chosen variable, and track signs — in service of ideas that happen to govern crashes, rockets, and pool tables.
Ready for the full picture? Head back to Physics or read the Student Support page for help getting unstuck.
Physics · enrichment — algebra-based mechanics offered for curiosity and cross-training. Not part of a TEKS-graded mathematics course and not scored toward any STAAR EOC.