Momentum and collisions formulas

Momentum = mass × velocity. Sign = direction: pick one way as +, the other way is −. Total momentum: add up every object's mv (with signs). 2-D crash at right angles, stuck together: type p as √(pₓ²+pᵧ²), e.g. √((1200*20)²+(1500*15)²), and m as m₁+m₂. Then v = speed after. Direction after: θ = tan⁻¹(pᵧ/pₓ); add 180° if pₓ < 0.

Any collision or explosion: total momentum before = after. Pick a + direction; moving the other way is negative. At rest = 0. Explosion or recoil from rest (gun, skaters push apart): v₁ = 0 and v₂ = 0. Moving together at first: type that same v in v₁ and v₂. 2-D: use it once with x parts (e.g. 5*cos(30)), once with y parts (5*sin(30)). Speed = √(vₓ²+vᵧ²), angle = tan⁻¹(vᵧ/vₓ) (+180° if vₓ<0).

Objects stick together and move as one (perfectly inelastic). Target at rest: v₂ = 0. Ballistic pendulum (block swings up h): type v' = √(2*9.8*h), solve for v₁. Know the swing angle instead? h = L(1 − cos θ). Target at rest: KE lost = ½m₁v₁² × m₂/(m₁+m₂). Other KE: K = ½mv² (Work & energy) before and after. 2-D: see the tip on p = mv.

Elastic = they bounce apart and no KE is lost (1-D). Target at rest: v₂ = 0. For v₂': swap the objects. Type object 2's mass and velocity in the 1 boxes, object 1's in the 2 boxes. The answer is v₂'. Equal masses: they just trade velocities. Check: v₁ − v₂ = v₂' − v₁'.

Average force that changes velocity in time Δt (bat, airbag, catching). Bounces back: v gets the opposite sign of v₀. Stops: v = 0. Angled hit: use only the part into the wall, e.g. v₀ = 20*cos(30), v = −20*cos(30). Dropped from h, up is +: v₀ = −√(2*9.8*h). Bounces up to h₂: v = √(2*9.8*h₂). F is the net force. Floor force on someone landing = F + mg. Hose or thrust: m = kg per second, Δt = 1.000 (not 1).

Impulse = change in momentum (J = Δp). Signs matter! Ball hits a wall at 20 m/s and bounces back at 15 m/s: v₀ = 20, v = −15. Stops: v = 0. Starts at rest: v₀ = 0. Angled hit: use only the part into the wall, e.g. v₀ = 20*cos(30), v = −20*cos(30). The other object gets −J (same size, opposite way).

Impulse = average force × time it pushes. Force vs time graph: J = area under it. Triangle spike: type F as half the peak (e.g. 1500/2). J points the same way as the force. Know the mass and speeds instead? Use J = m(v − v₀) (one up in the list).

2 objects: leave m₃ and x₃ at 0. 3 objects: type over them. Put x = 0 anywhere you like (often on one object). 2-D: use it once for x, once for y. Velocity of center of mass: type velocities in the x boxes; answer = vcm (units m/s, not m). Uniform rod, ball or plate: its mass sits at its middle. Person walks L on a boat: m₁ = person, x₁ = L, m₂ = boat, x₂ = 0. Answer xcm = how far the boat slides back.

Speed a rocket gains by burning fuel. In space: leave Δt = 0. Straight-up launch near Earth: Δt = burn time. mᵢ = full mass at the start. m = mass after the burn (mᵢ − fuel used). u = exhaust speed (how fast the gas leaves the rocket). Thrust force = u × kg of fuel burned per second. Acceleration = thrust / mass − 9.8 (no − 9.8 in space).