# What is the average speed of an object that is moving at 8 ms^-1 at t=0 and accelerates at a rate of a(t) =t-4 on t in [0,3]?

Jan 14, 2016

The average speed of an object is given by the area under its acceleration curve, and in this case is $0.5 m {s}^{-} 1$.

#### Explanation:

Speed is given by the area under the acceleration vs time curve for an object, and the mathematical tool we use to find the area under curves is integration. To answer this question we need to integrate $a \left(t\right) = t - 4$ between the limits $t = 0 s$ and $t = 3 s$, and know that the initial value for the speed is $8 m {s}^{-} 1$.

Using 's' for the average speed:

$s = {\int}_{0}^{3} \left(t - 4. \mathrm{dt}\right) = {\left[\left({t}^{2} / 2\right) - 4 t\right]}_{0}^{3} = \left(\left(\frac{9}{2} - 12\right) - \left(0 - 0\right)\right) = - 7.5 m {s}^{-} 1$

We can simply add this to the initial speed of $8 m {s}^{-} 1$ to give an average speed between 0 and 3 seconds of $0.5 m {s}^{-} 1$.