Consider first the velocity-time graph of an object moving for t seconds with constant acceleration a units
Suppose that at the beginning of the time interval the velocity is a units and at the end it is v units.
The velocity increases by a units each second so after t seconds the increase in velocity is at units.
v = a + at
This formula can be used in solving a problem on motion with uniform acceleration, provided that three out of the four quantities u, v, a and t are known, so that the fourth quantity can be calculated. If this is not the case we need another relationship; this can be found if we consider the displacement, s units, of the object from its starting point after t seconds.
The area under the velocity-time graph represents thr displacement. This is the area of a trapezium and is '-, ( u + v ) x t.
s = 1/2(u+v)t
Now we have a formula to use in those problems where three out of the four-quantity u, v, s and t are known.
There are however other possibilities. We could, for example, be given information on the values of u, a and t and have to find the displacement. Neither of the formulae found above link these four quantities but we can use them to deduce another relationship if we eliminate r.
From [I] v = u + at
Substituting in [2] gives s = 1/2 ( u + u + at )t,
--------[3]
In a similar way eliminating a gives
---------[4]
Lastly, a link between u, v, a and s can be found if t is eliminated from [1] and [2].
From [1]
Substituting in[2] gives s =
With these formulae established, we are in a position to tackle, by calculation, any problem on motion with constant acceleration.
Each formula contains four quantities, but not the fifth, from u, v, a, s and t so it is easy to identify the one to use by noting which quantity is not involved.
However always remember that, as we have already seen, many problems can be solved quickly and easily from a velocity-time graph using only the two basic facts that the gradient gives the acceleration and the area under the graph gives the displacement. Solution by graphical methods should not be neglected because calculation is now an alternative. In fact, even when using the formulae, a velocity-time sketch graph often makes the solution clearer.
