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
-
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 ).
-
Momentum Conservation:
- Even though energy is not conserved in this inelastic collision, momentum remains conserved, which is the key principle used in the derivation.
-
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