Motion and forces

A scalar has magnitude only: distance, speed, mass and elapsed time are examples. A vector has magnitude and direction: displacement, velocity, acceleration, momentum and force are examples. Choose axes before calculating. On an east–west line, east may be positive and west negative. The symbol Δ means final minus initial, so displacement s = Δx = x_f − x_i. A negative displacement describes direction rather than negative distance travelled. Average velocity is Δx/Δt; average speed uses the entire path length instead.

Positions must share the same origin.

Add vectors head to tail or add their signed components. Subtraction means adding the opposite vector. For a vector A at angle θ above the positive horizontal axis, A_x = A cos θ and A_y = A sin θ. Combine components separately, then use magnitude √(A_x² + A_y²) and a direction consistent with the signs. Pythagoras applies to perpendicular components, not arbitrary pairs of vectors.

For constant acceleration, a = Δv/Δt, v_f = v_i + aΔt, s = v_iΔt + ½aΔt² and v_f² = v_i² + 2as. Select an equation containing the known quantities and requested unknown. A trolley initially moving at 2.0 m/s with acceleration 2.0 m/s² for 3.0 s reaches 8.0 m/s and travels 15 m. These equations do not assume initial rest.

A displacement–time graph has velocity as its slope; curvature reveals changing velocity. A velocity–time graph has acceleration as its slope and signed displacement as its area. An acceleration–time graph has change in velocity as its signed area. Areas below the time axis are negative. With upward positive, near Earth's surface a = −9.8 m/s² when air resistance is negligible. At the top of a throw velocity is zero momentarily, but acceleration remains downward.

A ball dropped 5.0 m reaches 9.9 m/s downward, using v_f² = 2as with acceleration and displacement both negative.