Here is the question:
If the velocity function for a projectile is v(t) = 4 cos ( t), which of the following is a possible corresponding position function?
Here is the answer:
Could someone explain this to me? Thanks
Hello vease1999$\displaystyle v = \frac{ds}{dt}=4\cos(\pi t)$
$\displaystyle \Rightarrow s = \int4\cos(\pi t)\,dt$$\displaystyle =\frac{4}{\pi}\sin(\pi t) + c$So if the initial conditions are such that $\displaystyle c = \frac{8}{\pi}$, the solution would be:$\displaystyle s = \frac{8+4\sin(\pi t)}{\pi}$Grandad
Hello vease1999
Velocity is the rate of change of position.$\displaystyle v = \frac{ds}{dt}=4\cos(\pi t)$So we integrate to find s.$\displaystyle \Rightarrow s = \int4\cos(\pi t)\,dt$Do you know how to integrate the cosine function?c can have any constant value; for example $\displaystyle \color{red}\frac{8}{\pi}$.$\displaystyle =\frac{4}{\pi}\sin(\pi t) + c$
So if the initial conditions are such that $\displaystyle c = \frac{8}{\pi}$, the solution would be:Grandad$\displaystyle s = \frac{8+4\sin(\pi t)}{\pi}$