constant acceleration equations explained

Change the acceleration a with the slider and the particle's positions at equal time intervals change, together with a blue arrow for its velocity at each moment. Watch the arrows: from one snapshot to the next they grow by the same amount, a. The snapshots are equally spaced in time, so the velocity gains a every second, which is exactly what constant acceleration means. It is not the velocity that stays constant but the amount it grows, so the gold dots spread farther apart toward the right. Raise a and the arrows grow more steeply and the particle travels much farther; set a to 0 and all the arrows match, which is the uniform motion of the last lesson.

With constant acceleration, the velocity-time graph is a straight line. Time t runs along the horizontal axis and velocity v up the vertical, and dragging the time slider moves the point up the blue line. Its height at the start, t = 0, is the initial velocity v0, marked by a short tick on the left axis. As time passes the velocity gains a each second, so the slope of the line is a, and in symbols v = v0 + at: start at v0 and add the same amount every second, nothing more to memorize. The line is fixed by two numbers, the starting value v0 and the slope a. With a positive slope, as in the figure, it climbs to the right and the particle speeds up; with a negative a, a deceleration, it falls to the right and the particle slows down. In the last lesson velocity was a slope; here that slope never changes, so velocity itself becomes a straight line.

As the last lesson showed, position is the area under the velocity graph, and with constant acceleration the velocity is a straight line, so that area is a trapezoid. Move the time slider and the shaded area builds to the right while the position reached so far appears in the readout. The trapezoid splits in two: the blue rectangle is the distance v0t the particle would have covered at its initial speed, and the orange triangle on top is the extra distance ½at² that the acceleration adds. Together they give x = x0 + v0t + ½at²: the starting position x0, plus the distance uniform motion would cover, plus what the acceleration contributes. With a at 0 the triangle vanishes, only the rectangle remains, and x = x0 + v0t is plain uniform motion.

Sometimes the time does not matter: you want to know how far the particle must go to reach a certain speed, or how fast it will be after a certain distance. Then a formula without time is handy. Take v = v0 + at and the x formula, eliminate the time t they share, and out comes v² = v0² + 2a·Δx. On a graph, v² is an orange straight line in the distance traveled Δx, starting from the height v0² marked on the left axis, with slope 2a. Drag the distance slider and the orange dot climbs the line while the readout gives the speed. Double the distance and the gain in v² doubles too; the key point is that the square of the speed, not the speed itself, is proportional to distance. That is why a car's braking distance becomes four times longer when its speed doubles.

The most famous case of constant acceleration is gravity. Ignoring air resistance, every falling object, heavy or light, is pulled downward with the same acceleration g, about 9.8 m/s². Switch with the buttons between dropping, throwing down, and throwing up: taking up as positive, the acceleration is always −g, and only the initial velocity v0 changes. So the three velocity-time lines start at different heights but share one slope; they are parallel. The chosen case is the thick blue line and the others stay faint. When the object is thrown up, the moment its velocity reaches 0 is the highest point, marked by the gold dot, and even there the acceleration is still −g: gravity does not pause when the object does. In the end, free fall is just constant acceleration with a = g directed downward, so all the earlier formulas apply unchanged.