Question

An object has a uniform acceleration a. After a time t its final velocity is v.
a) Sketch a graph of velocity against time for this object.
b) Show that the displacement of the object in this time is given by:
s= vt - (1\2) at^2

Answer 1

A velocity time graph has a gradient which is equal to its acceleration. This is because the differential of velocity:
\[\frac{ dv }{ dt }=a\]
Is acceleration. By uniform acceleration I assume you mean a constant acceleration, in this case constant acceleration is given by the straight line on the vt graph. If the object was not accelerating, its velocity would be unchanging (unless it changed direction, but that's irrelevant in this case). An object with no acceleration on a vt graph would be a flat horizontal line, in other words a gradient of zero.

Answer 2

For part 2, do you need to derive using the equations of motion or show from a velocity time graph? I'm assuming from the graph judging by the previous question, the area beneath a vt graph is displacement. Try and play around with the area using ideas of areas of triangles and rectangles. If I'm honest I'm not too sure myself so I'll have a look

Answer 3

Use this website to try and derive the expression
http://www.nuffieldfoundation.org/practical-physics/using-speed-time-graphs-find-equation
Requires a bit of substitution but it's very much possible