Answer
(a) $3T$
(b) $v\sqrt{2}$
Work Step by Step
(a) We can find the required time as follows:
$P=\frac{W}{t}$
But $W=\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2$
$\implies P=\frac{\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2}{T}$
$\implies P=\frac{\frac{1}{2}mv_f^2-\frac{1}{2}m(0m/s)^2}{T}$
$P=\frac{1}{2}\frac{mv^2}{T}$........eq(1)
As given that the power output of the car from $v_i=vm/s$ to $v_f=2vm/s$
$\implies P=\frac{\frac{1}{2}m(2v)^2-\frac{1}{2}mv^2}{t}$
$\implies P=\frac{1}{2}m\frac{3v^2}{t}$.....eq(2)
Given that the power output of the car is the same
$\implies \frac{1}{2}\frac{mv^2}{T}=\frac{1}{2}m\frac{3v^2}{t}$
$\implies t=3T$
(b) We can find the car final speed as follows:
From eq(1)
$v=\sqrt{\frac{2PT}{m}}$
$\implies v_{2T}=\sqrt{\frac{2P(2T)}{m}}$
$\implies v_{2T}=\sqrt{2}\sqrt{\frac{2PT}{m}}$
$\implies v_{2T}=v\sqrt{2}$
