Motion and projectiles formulas

Use when time is in the problem (steady a). Up/right is +. Free fall: a = −9.8. Stops, or top of a throw: v = 0. Lands at launch height: v = −v₀ gives the flight time. Projectile: do x and y apart. y: a = −9.8, v₀ = speed×sin(θ), like 22×sin(40). x: a = 0. Straight-line a-t graph: a = (a₁+a₂)/2, t = t₂−t₁. v(t) = 3 + 4t: v₀ = 3, a = 4. x(t) = A + Bt + Ct²: v₀ = B, a = 2C. Sliding to a stop on friction: a = −μ×9.8.

Average velocity, or steady speed (a = 0). Can be −. Δx = end − start position. t = end − start time. Average SPEED: Δx = total distance. Steady a (speed-up or braking): vₐ = (v₀+v)/2. Legs: Δx = sum of v×t (west −), t = total. Projectile sideways: vₐ = speed×cos(θ). River crossing: Δx = width, vₐ = speed across. Circle speed: Δx = 2πr, t = 1/f (f = rpm/60). mph with hours is OK. km/h to m/s: type 90/3.6.

Use when Δx and t are in the problem. Empty t: earliest first. Up/right is +. Free fall: a = −9.8. Up/down: Δx is Δy. Dropped, or thrown level: y part v₀ = 0, Δy = −height. Sideways: a = 0, v₀ = speed (type it +). At an angle: y part v₀ = speed×sin(θ), like 22×sin(40). Lands at launch height (Δy = 0): flight time is the "or" answer. Catch-up: Δx = gap, v₀ = v₁−v₂, a = a₁−a₂ (1 = chaser). km/h to m/s: type 150/3.6.

Use when there is no time in the problem. Up/right is +. Free fall: a = −9.8. Top of a throw: v = 0. Leave Δx empty to get the max height. Projectile max height: y part v₀ = speed×sin(θ), like 22×sin(40). v or v₀ comes out + or −: + first. Moving down or left: use the − one. From rest: v₀ = 0. Stops: v = 0. Friction: a = −μ×9.8.

Only when it lands at the height it was launched from. R = sideways distance. θ = angle above level. Empty θ: 2 angles work (they add to 90°). Longest range: θ = 45°. Max height: v² formula, v₀ = speed×sin(θ), v = 0, a = −9.8. Time in air: Δx formula, Δx = 0, a = −9.8, v₀ = speed×sin(θ); take the "or" answer. Not level: do x and y apart (x: a = 0, y: a = −9.8). Speed + angle from x and y parts: Measurement & vectors.