The correct answer is option 2: the kinetic energy will increase by four times.

To understand why this is the case, we need to consider the relationship between momentum and kinetic energy. Linear momentum is the product of an object`s mass and its velocity. The formula for linear momentum is p = mv, where p is the momentum, m is the mass, and v is the velocity.

On the other hand, kinetic energy is the energy possessed by a moving object due to its motion. The formula for kinetic energy is KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.

When the linear momentum is doubled, it means that the product of mass and velocity (mv) is two times its original value. However, kinetic energy is not directly proportional to momentum. It depends on the square of velocity.

Since the velocity is an important factor in kinetic energy, when the linear momentum is doubled, we can assume that the velocity will also get doubled to maintain the same proportionality. Therefore, if the velocity is doubled, the kinetic energy will increase by four times, as indicated in option 2.