Velocity-Time Law

Multiply the acceleration by the time and add the initial velocity. Example: a train travels at v₀ = 10 m/s and accelerates for 8 s at a = 1.5 m/s²: v = 1.5 × 8 + 10 = 22 m/s, which is about 79 km/h. All quantities belong in SI units; divide a speed in km/h by 3.6 before substituting. Intuitively the formula says: the acceleration states how many m/s the body gains per second. At a = 1.5 m/s² it thus gets 1.5 m/s faster every second. When braking, insert a as negative and v decreases accordingly.

It means the velocity increases by 1 m/s every second; after three seconds the body is 3 m/s faster. The double unit m/s² is best read as "(m/s) per s". For scale: a brisk small car manages about 3 m/s² when pulling away, emergency braking is 8 to 10 m/s², free fall is 9.81 m/s², and a Formula 1 car brakes at up to 50 m/s² (about 5g). Acceleration is not limited to speeding up: physically every change of velocity is an acceleration, including braking (negative sign) and even pure changes of direction in circular motion.

First subtract v₀ on both sides, then divide by a: t = (v − v₀)/a. The numerator is the change in velocity, not the final velocity, which is the most common mistake. Example: a car accelerates at a = 2.5 m/s² from 0 to 100 km/h (27.8 m/s): t = (27.8 − 0)/2.5 ≈ 11.1 s. Slowing from 26 m/s to 20 m/s at a = −2 m/s²: t = (20 − 26)/(−2) = 3 s; two negative signs cancel and the time is positive as expected. If a negative time comes out, signs or start and end values are swapped. A unit check (m/s divided by m/s² gives s) secures the result.

Velocity says how fast the position changes (m/s); acceleration says how fast the velocity changes (m/s²). Both can independently be large, small or zero: an aircraft in cruise is very fast but unaccelerated (a = 0). A ball at the top of a vertical throw momentarily has v = 0 yet is fully accelerated at g = 9.81 m/s², otherwise it would stay up there. The directions need not agree either: when braking, a points against the motion. This distinction is the core of many conceptual questions; anyone who equates v and a fails exactly these tasks.

Remember two rules: slope and area. In the s-t diagram the slope is the velocity; a steeper curve means faster, a parabola indicates acceleration. In the v-t diagram the slope is the acceleration: a line with positive slope means constant a, a horizontal line uniform motion. Additionally, the area under the v-t curve is the distance travelled; for the trapezoid made of the v₀ rectangle and the acceleration triangle this gives exactly s = v₀t + ½at². Exam problems like to combine segments: first accelerate, then drive at constant speed, then brake. Split the diagram into these phases and compute the areas separately.