If the speed of a body is increased four times, how will it's kinetic energy be affected?
Figuring out this answer just depends upon knowing that kinetic energy is equivalent to half the mass, times the velocity squared: KE = .5mv^2
So if we set the velocity equal to 4v, we can extrapolate the result simply by looking at how 4v will look when squared;
4v^2 = (4v) (4v) = 16v^2
The only difference compared to the normal value of v is that 16 in front; so the kinetic energy will be 16 times greater.
We can observe this with an example;
let m = 10kg
let v = 10m/s
KE = .5mv^2
KE = .5(10)(10^2)
KE = 500
If quadrupling the speed produces a kinetic energy 16 times greater, then the KE of 4v should be 500(16) = 8000
KE = .5(10)(40^2)
KE = 8000
Conceptually this helps to explain why more massive objects almost always "win" in a collision with less massive objects at comparable speeds, and why it's easier to fire less massive objects at high speeds when it comes to ballistic weapons and rockets. It's important to remember that the velocity figures significantly more into the evaluation of the overall kinetic energy, at least in part because the definition of kinetic energy itself is based on motion.