Elastic Collision Analysis

Elastic Collision Problem

Problem Statement

  • A 0.280 kg croquet ball makes an elastic head-on collision with a second ball that is initially at rest.
  • Post-collision, the second ball moves with half the original speed of the first ball.

Given Data

  • Mass of first ball (m1): 0.280 kg
  • Initial speed of first ball (u1): To be determined.
  • Second ball (m2): Mass to be determined.
  • Final speed of second ball (v2): Half of the original speed of the first ball, i.e., v2 = \frac{1}{2} u1

Goals

  • (a) Determine the mass of the second ball, m2.
  • (b) Calculate the fraction of the original kinetic energy that gets transferred to the second ball, expressed as \frac{AKE}{KE}.

Solution Approach

(a) Finding the mass of the second ball

  1. Conservation of Momentum:

    • For elastic collisions, momentum before collision = momentum after collision.
    • The equation can be expressed as:
      m1 \cdot u1 + m2 \cdot 0 = m1 \cdot v1 + m2 \cdot v_2
    • Here, v1 is the final speed of the first ball (to be determined).

  2. Using the relationship between speeds after the collision:

    • Since the second ball moves with half the original speed of the first ball, we let v2 be defined as:
      v2 = \frac{1}{2} u1
    • Substitute v2 and express the momentum conservation equation as:
      0.280 \cdot u1 = 0.280 \cdot v1 + m2 \cdot \frac{1}{2} u1

  3. Eliminating variables and simplifying:

    • To find m2, we need an additional equation. We can use the conservation of kinetic energy simultaneously because it is an elastic collision.

(b) Fraction of Original Kinetic Energy Transferred

  1. Kinetic Energy Before Collision:

    • The kinetic energy (KE) of the first ball before the collision can be expressed as:
      KE{initial} = \frac{1}{2} m1 u1^2 = \frac{1}{2} \cdot 0.280 \cdot u1^2

  2. Kinetic Energy After Collision:

    • The kinetic energy of the second ball after the collision is:
      KE{final} = \frac{1}{2} m2 v_2^2
    • Substituting v2 gives:
      KE{final} = \frac{1}{2} m2 \left(\frac{1}{2} u1\right)^2 = \frac{1}{2} m2 \cdot \frac{1}{4} u1^2 = \frac{1}{8} m2 u_1^2

  3. Finding the fraction of kinetic energy transferred:

    • The fraction of kinetic energy transferred to the second ball can be expressed as:
      \frac{KE{final}}{KE{initial}} = \frac{\frac{1}{8} m2 u1^2}{\frac{1}{2} \cdot 0.280 \cdot u1^2} = \frac{\frac{1}{8} m2}{\frac{1}{2} \cdot 0.280}

  4. Simplifying the fraction:

    • This can be further simplified by substituting known values and solving for the fraction transferred, once m2 is known.