By what factor does the kinetic energy of a car change, if an increase in speed doubles the momentum of the car?

The kinetic energy and momentum of a body are given as:

Kinetic energy or K.E. = 1/2 mv^2

and momentum, P = m x v

where, m is the mass of the body and v is its velocity.

Let us say the car has a mass of 'm' and a velocity of 'v' initially.

If due to speed change, the momentum is twice the original value, then

P' = m x v' = 2 P = 2 m x v

or, v' = 2 v

Note that mass of the car remains the same and hence only the velocity is double the original value.

The new kinetic energy will be:

K.E.' = 1/2 m v'^2 = 1/2 m (2v)^2 = 4 [ 1/2 mv^2] = 4 K.E.

The new kinetic energy is thus 4 times the original kinetic energy. 

Hope this helps.