If our speed increases three times, our kinetic energy increases by a factor of nine, that is, our kinetic energy increases nine times.
The kinetic energy is a form, of mechanical energy and it is associated with the movement of the bodies. All body with mass m, which moves with a speed v, has associated a kinetic energy given by the following expression:
Ek = (mv^2)/2
m, is the mass of the body.
v, is the velocity of the body.
The kinetic energy Ek1, for the initial velocity v1 is:
Ek1 = (mv1^2)/2
As we see, the kinetic energy is proportional to the square of the speed. If the initial speed v1 increased three times, the final speed v2, will be:
v2 = 3v1
Then the kinetic energy Ek2, for the speed v2, will be:
Ek2 = (mv2^2)/2 = [m(3v1)^2]/2
Ek2 = 9(mv1^2)/2 = 9(Ek1)
So, when the speed increases three times, the kinetic energy increases nine times.