If the linear momentum of a moving object is doubled (mass unchanged), its kinetic energy:

Kinetic energy (KE) is the energy possessed by an object due to its motion. It is given by:

KE = ½mv²

where m is the mass and v is the velocity of the object.

Linear momentum (p) is the product of mass and velocity:

• p = mv
• The relation between kinetic energy and momentum can be found by substituting v = p/m into the formula for kinetic energy:
• KE = ½m(p/m)² = p²/2m
• This shows that, for a given mass, kinetic energy is directly proportional to the square of the momentum (KE ∝ p²).

So, if the momentum is doubled (p becomes 2p) while the mass stays the same, the kinetic energy becomes (2p)²/2m = 4p²/2m = 4 × (p²/2m). In other words, the kinetic energy becomes four times its original value.

Since the mass is unchanged, doubling the momentum means doubling the velocity. And since KE depends on the square of velocity, doubling the velocity also gives four times the kinetic energy. Both approaches give the same answer.

Similarly:

• If the velocity is tripled, the kinetic energy becomes nine times.
• If the kinetic energy is increased four times, the momentum doubles (p ∝ √KE).
• If two objects have the same kinetic energy, the heavier object has more momentum (because p = √(2m × KE)).
• If two objects have the same momentum, the lighter object has more kinetic energy (because KE = p²/2m).

This explains why speed is so dangerous in road accidents: a car moving at twice the speed has four times the kinetic energy, needs about four times the distance to stop with the same braking force, and causes much greater damage in a collision.

Units: kinetic energy is measured in joules (J), and momentum in kilogram metre per second (kg m/s). Kinetic energy is a scalar quantity, while momentum is a vector.

Why not the other options?

• ✗Doubles: incorrect, because KE depends on the square of momentum.
• ✗Remains the same: incorrect.
• ✗Becomes half: incorrect.