In an elastic collision, both momentum and kinetic energy are conserved. We can use these conservation laws to solve for the velocities after the collision.
Let's denote:
- \( m_1 = 5 , \text{kg} \) (mass of the first cart)
- \( v_1 = 0 , \text{m/s} \) (initial velocity of the first cart)
- \( m_2 = 1 , \text{kg} \) (mass of the second cart)
- \( v_2 = 3 , \text{m/s} \) (initial velocity of the second cart)
- \( v_1' \) (final velocity of the first cart)
- \( v_2' \) (final velocity of the second cart)
Step 1: Conservation of Momentum
The equation for conservation of momentum is: \[ m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2' \] Substituting the values we know: \[ 5 \cdot 0 + 1 \cdot 3 = 5 v_1' + 1 v_2' \] This simplifies to: \[ 3 = 5 v_1' + v_2' \quad \text{(1)} \]
Step 2: Conservation of Kinetic Energy
The equation for conservation of kinetic energy is: \[ \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 {v_1'}^2 + \frac{1}{2} m_2 {v_2'}^2 \] Substituting the known values: \[ \frac{1}{2} \cdot 5 \cdot 0^2 + \frac{1}{2} \cdot 1 \cdot 3^2 = \frac{1}{2} \cdot 5 \cdot {v_1'}^2 + \frac{1}{2} \cdot 1 \cdot {v_2'}^2 \] This simplifies to: \[ \frac{1}{2} \cdot 1 \cdot 9 = \frac{1}{2} \cdot 5 \cdot {v_1'}^2 + \frac{1}{2} \cdot 1 \cdot {v_2'}^2 \] \[ 4.5 = \frac{5}{2} {v_1'}^2 + \frac{1}{2} {v_2'}^2 \quad \text{(2)} \]
Step 3: Solve the system of equations
From equation (1): \[ v_2' = 3 - 5 v_1' \quad \text{(3)} \] Substituting equation (3) into equation (2): \[ 4.5 = \frac{5}{2} {v_1'}^2 + \frac{1}{2} (3 - 5 v_1')^2 \] Expanding this: \[ 4.5 = \frac{5}{2} {v_1'}^2 + \frac{1}{2} (9 - 30 v_1' + 25 {v_1'}^2) \] \[ 4.5 = \frac{5}{2} {v_1'}^2 + \frac{1}{2} \cdot 9 - 15 v_1' + \frac{25}{2} {v_1'}^2 \] Combining like terms: \[ 4.5 = \left( \frac{5}{2} + \frac{25}{2} \right) {v_1'}^2 - 15 v_1' + \frac{9}{2} \] \[ 4.5 = 15 {v_1'}^2 - 15 v_1' + 4.5 \] Subtract \( 4.5 \) from both sides: \[ 0 = 15 {v_1'}^2 - 15 v_1' \] Factoring out \( 15 v_1' \): \[ 0 = 15 v_1' (v_1' - 1) \] This gives us the solutions: \[ v_1' = 0 \quad \text{or} \quad v_1' = 1 , \text{m/s} \]
Conclusion
The first cart (5 kg) has a final velocity of \( \mathbf{1 , m/s} \) after the collision.