Question

A car accelerates with the acceleration a(s) = k · s^n m/s^2, where s is distance in meters, k = 1.8 and n = 0.725. (Unless otherwise stated, assume that the car has a  negligible initial speed, which only means that the answer does not become trivial  and that the car starts at s = 0.)
How far has the car driven when it has reached a speed of 1.4 m/s?

Answer 1

$v^2=v_o^2+2as$

$v^2=2as$

$1.4^2=2*k*s^n*s$

$1.4^2=2\cdot \:1.8ss^{0.725}$

$1.4^2=3.6s^{0.725}s$

$\mathrm{Use\:the\:following\:exponent\:property}:\quad a^n=\left(a^{\frac{1}{m}}\right)^{\left(n\cdot \:m\right)}$

$1.4^2=3.6u^{29}u^{40}$

$s=\left(\frac{1.96}{3.6}\right)^{\frac{40}{69}}$

$1.4^2=3.6\left(s^{\frac{1}{40}}\right)^{29}\left(s^{\frac{1}{40}}\right)^{40}$

$s=\left(\frac{1.96}{3.6}\right)^{\frac{40}{69}}=s=0.70295 m$