I'm working through some exercises and have again come across one that is giving me some trouble. The topic is calculating velocity and acceleration when time varies. Here it is:
A body moves in such a way that the space described in the time $t$ from starting is given by $s$ = $t^n$, where $n$ is a constant. Find the value of $n$ when the velocity is doubled from the 5th to the 10th second; find it also when the velocity is numerically equal to the acceleration at the end of the 10th second.
I understand the velocity of this to be $v$ = $nt^{n-1}$ and the acceleration to be $a$ = $(n - 1)nt^{n - 2}$.
The other exercises in this chapter barely compare to this one. This is the last exercise of the chapter and as expected is the most difficult. Any help on this is truly appreciated.
For the first, just take ratios. $v(5) = n \cdot 5^{n-1}$; $v(10)=n \cdot 10^{n-1}$. Then
$$\frac{v(1)}{v(5)} = \left ( \frac{10}{5}\right)^{n-1} = 2$$
Clearly, $n=2$. For the second, it is similar:
$$v(10) = a(10) \implies n \cdot (10)^{n-1} = n(n-1) \cdot (10)^{n-2}$$
or $10 = (n-1)$.