ALAMSCIENCE

One graph,
all of SUVAT.

We think in graphs. On a straight v-t line, the slope is the acceleration and the area under it is the displacement. The five equations are not five things to memorise: you read them all off the same graph.

Velocity-time graph: starts at u with slope a and reaches v at time t A straight line rises from the y-intercept u and reaches v at time t. The trapezium under the line splits into a rectangle of height u and a triangle of height v minus u. velocity v / m s⁻¹ v-t graph: slope = a, area = s t rectangle ut triangle ½at² u v t slope = a

One trapezium, split three ways, gives the three displacement equations.

Physics simulations

Slope is acceleration. Area is displacement.

The three displacement equations are the same trapezium, with its area worked out three ways. Forgot one? Look back at the graph and derive it in ten seconds.

0
Definition of slopeThe line rises from u to v in time t, so its slope is a. This is y = mx + c from S3 maths: v is y, t is x, the slope m is a and the y-intercept c is u.
v = u + at
1
Trapezium = rectangle + triangleThe rectangle below is u high and t wide; the triangle on top is at high.
s = ut + ½at²
2
Trapezium = big rectangle − triangleTake v as the height, then subtract the triangle on top. Nobody teaches this one, but it is just as correct.
s = vt − ½at²
3
Primary-school trapezium area: (top + bottom) × height ÷ 2Top u, bottom v, height t. (u + v) / 2 is the average velocity v̄, so this is simply average velocity × time.
s = (u + v)t / 2 = v̄t
4
The fifth: take the energy routeThe change in kinetic energy equals the work done, and work is force × displacement. Cancel the mass m and only velocity, acceleration and displacement remain. Eliminating t from the first four gives the same equation, so the two routes agree: physics is self-consistent.
½mv² − ½mu² = W = Fs = mas　→　v² − u² = 2as
v² − u² = 2as

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