Question

If a moving car speeds up until it is going twice as fast, how much kinetic energy doe s it have compared with its initial kinetic enegy

Answer 1

The kinetic energy of an object of mass m and velocity v is given by
$K= \frac{1}{2} mv^2$

Let's call $v_i$ the initial speed of the car, so that its initial kinetic energy is
$K_i = \frac{1}{2} mv_i^2$
where m is the mass of the car. 

The problem says that the car speeds up until its velocity is twice the original one, so 
$v_f = 2 v_i$
and by using the new velocity we can calculate the final kinetic energy of the car
$K_f = \frac{1}{2} mv_f^2 = \frac{1}{2}m (2 v_i)^2 = 4 ( \frac{1}{2} mv_i^2)=4 K_i$
so, if the velocity of the car is doubled, the new kinetic energy is 4 times the initial kinetic energy.