Elastic Collision: Mass m2 Velocity & Momentum of m1

[chynawok]

Suppose that a particle of mass m1 approaches a stationary mass m2 and that m2 > m1.
a) Describe the velocity of m2 after an elastic collision--that is, one in which both momentum and kinetic energy are conserved. Justify the answer mathematically.
b) What is the approximate momentum of m1 after the collision?

 

[Pyrrhus]

Read the rules, you need to show some work, start setting up the equations of conservation.

 

[thepatientmental]

a) In an elastic collision, both momentum and kinetic energy are conserved. This means that the total momentum and total kinetic energy of the system before and after the collision are equal. In this scenario, the initial momentum of m1 is given by its mass (m1) multiplied by its initial velocity (v1). The initial momentum of m2 is zero as it is stationary. After the collision, the final momentum of m1 can be calculated by multiplying its mass (m1) by its final velocity (v1'). Similarly, the final momentum of m2 can be calculated by multiplying its mass (m2) by its final velocity (v2'). The equation for conservation of momentum can be written as m1v1 + m2v2 = m1v1' + m2v2'. Since m2 > m1, it can be assumed that m2 will have a lower velocity after the collision compared to m1. This is because the total momentum of the system must remain constant. Therefore, the velocity of m2 after the collision will be less than the velocity of m1 before the collision.

b) The approximate momentum of m1 after the collision can be calculated using the equation for conservation of momentum mentioned above. Rearranging the equation, we get m1v1' = m1v1 + m2v2 - m2v2'. Since m2 > m1, m2v2 - m2v2' will be a positive value, which means that m1v1' will be greater than m1v1. This indicates that the final velocity of m1 (v1') will be greater than its initial velocity (v1). Therefore, the approximate momentum of m1 after the collision will be greater than its initial momentum. This can also be justified mathematically by considering the fact that kinetic energy is also conserved in an elastic collision. The equation for conservation of kinetic energy can be written as 1/2m1v1^2 = 1/2m1v1'^2. Since m1 is constant, this means that v1'^2 > v1^2, which further supports the idea that the final momentum of m1 will be greater than its initial momentum.