Let a body of mass m and velocity v is brought to test in time t,
Initial momentum = mv,
Final momentum = m× 0 = 0 ( as the ball is brought to test its velocity is zero),
Change in momentum = final momentum - initial momentum= 0 - mv,
Time in which ball is brought to rest = t,
Rate of change in momentum = - m v/t,
Force F = Rate of change of momentum =- mv/t,
Impulse = Force F × t = change in, momentum = -mv,
The only change is in the rate of change of momentum, which is doubled when the time is halved. Though impulse (=change in momentum) does not change, the force required is doubled.

Changes in Momentum and Rate of Change of Momentum when the time taken to bring a ball to rest from a certain velocity v is reduced to half.

When an object is brought to rest from a certain velocity, a change in momentum occurs. The momentum of an object is the product of its mass and velocity. The rate of change of momentum is the force acting on the object. According to Newton's Second Law of Motion, the force acting on an object is directly proportional to the rate of change of momentum.

When the time taken to bring a ball to rest from a certain velocity v is reduced to half, the following changes occur:

Initial and final momentum:

The initial momentum of the ball is given by the product of its mass and velocity. If the velocity is v, then the initial momentum is mv. When the ball is brought to rest, the final momentum is zero.

If the time taken to bring the ball to rest is reduced to half, the final velocity of the ball will be half of the initial velocity. This is because the acceleration of the ball is constant, and the distance covered by the ball is proportional to the square of the time taken. As a result, the final momentum of the ball will be half of the initial momentum.

Change of momentum:

The change in momentum of the ball is given by the difference between the final and initial momentum. If the initial momentum is mv and the final momentum is zero, then the change in momentum is -mv.

If the time taken to bring the ball to rest is reduced to half, the final momentum of the ball will be half of the initial momentum. As a result, the change in momentum of the ball will be -0.5mv.

Rate of change of momentum:

The rate of change of momentum is the force acting on the ball. According to Newton's Second Law of Motion, the force acting on an object is directly proportional to the rate of change of momentum.

If the time taken to bring the ball to rest is reduced to half, the change in momentum of the ball will be half of the initial change in momentum. As a result, the force acting on the ball will be half of the initial force.