Derive an equation for the speed vc of the dart sphere system

General
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derive an equation for the speed vc of the dart sphere system

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To derive the equation for the speed (v_c) of the dart-sphere system immediately after a dart’s collision with a sphere, we’ll use the law of conservation of linear momentum. Below is the derivation step-by-step:

1. Understanding the System

This problem involves a dart (mass ( m_d ), initial velocity ( v_d )) being fired at a stationary sphere (mass ( m_s )):

  • The dart sticks to the sphere upon collision, forming a composite system with mass ( m_d + m_s ).
  • This is an inelastic collision, where momentum is conserved but kinetic energy is not necessarily conserved.

We aim to derive an expression for the velocity ( v_c ), which is the velocity of the combined system (dart + sphere) immediately after the collision.

2. Applying Conservation of Linear Momentum

The total linear momentum before the collision equals the total linear momentum after the collision. Let’s write it mathematically:

[
m_d v_d + m_s v_s = (m_d + m_s) v_c
]

Breaking it Down:

  • Before the collision:
    • The dart’s momentum is ( m_d v_d ).
    • The sphere is stationary, so its momentum is ( m_s v_s = 0 ).
  • After the collision:
    • The combined system (dart + sphere) moves together with velocity ( v_c ), so its momentum is ( (m_d + m_s)v_c ).

Thus, the equation simplifies to:

[
m_d v_d = (m_d + m_s) v_c
]

3. Solving for ( v_c )

To find ( v_c ), isolate it on one side of the equation:

[
v_c = \frac{m_d v_d}{m_d + m_s}
]

4. Final Expression

The speed ( v_c ) of the dart-sphere system immediately after the collision is:

[
v_c = \frac{m_d v_d}{m_d + m_s}
]

Key Observations

  1. Dependence on Mass and Initial Velocity:

    • The larger the mass of the sphere (( m_s )), the smaller the resulting velocity ( v_c ).
    • The larger the dart’s initial velocity (( v_d )), the larger ( v_c ).

  2. Momentum Conservation:

    • Even though energy is not conserved in this inelastic collision, momentum remains conserved, which is the key principle used in the derivation.

  3. Practical Applications:

    • This result is commonly used in physics problems involving collisions, especially where two objects stick together post-collision (e.g., missile hitting a satellite, ball sticking to the ground).

Let me know if you need additional clarifications! @anonymous13