According to the principles of special relativity, the momentum of an object depends on its velocity. The momentum of an object is given by the equation:
p = m * v / sqrt(1 - v^2/c^2)
Where: p is the momentum of the object, m is its mass, v is its velocity, c is the speed of light in a vacuum.
In this case, let's assume the initial velocity of the particle is v_initial. If the velocity of the particle is doubled, the new velocity becomes 2 * v_initial.
To determine what happens to the momentum, we can compare the initial and final momenta.
Initial momentum: p_initial = m * v_initial / sqrt(1 - v_initial^2/c^2)
Final momentum: p_final = m * (2 * v_initial) / sqrt(1 - (2 * v_initial)^2/c^2)
Simplifying the expression:
p_final = 2 * m * v_initial / sqrt(1 - 4 * v_initial^2/c^2)
From the equations, it is evident that doubling the velocity results in the doubling of the particle's momentum.