Projectile Motion

Projectile motion is the motion of an object that is thrown into the air at an angle or launched horizontally, under the influence of gravity. In this case, a particle of mass m is projected with velocity v at an angle of 45 degrees with the horizontal.

Components of Velocity

When a particle is projected at an angle, its initial velocity can be resolved into two components: one along the horizontal direction (v_x) and one along the vertical direction (v_y). In this case, since the angle of projection is 45 degrees, both components will have the same magnitude, v/√2.

Momentum

Momentum is defined as the product of an object's mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction. The magnitude of momentum can be calculated using the equation p = mv, where p is the momentum, m is the mass, and v is the velocity.

Change in Momentum

The change in momentum (∆p) is the difference between the final momentum and the initial momentum. In this case, the particle lands on level ground, which means its final vertical velocity is 0. Therefore, the final momentum is only due to the horizontal component of velocity.

Calculating the Change in Momentum

To calculate the change in momentum, we need to find the final horizontal momentum and subtract the initial horizontal momentum. Since the particle lands at the same horizontal position from where it was launched, the change in horizontal momentum is equal to the initial horizontal momentum.

The initial horizontal momentum can be calculated using the equation p_x = m * v_x. As mentioned earlier, v_x = v/√2. Therefore, the initial horizontal momentum is m * (v/√2).

The change in momentum (∆p) is then 2 * m * (v/√2) = √2 * m * v.

Conclusion

In conclusion, when a particle of mass m is projected with velocity v at an angle of 45 degrees with the horizontal, the magnitude of the change in its momentum when it lands on level ground is √2 * m * v.