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

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

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