While carelessly riding his bike at 26.0 m/s east, Pablo crashes into a parked car with a mass of 1,780 kg. As a result, Pablo comes to a stop and the car moves. Pablo and his bike have a mass of 87.0 kg. Determine the velocity of the car after the collision.
The correct answer and explanation is:
To solve this problem, we can use the principle of conservation of momentum. The total momentum of a system remains constant if no external forces act on it. In this case, the system consists of Pablo and his bike and the car. Since the collision is described as inelastic (Pablo comes to a stop), the momentum before the collision must equal the momentum after the collision.
Step 1: Define the variables
- Mass of Pablo and his bike, m1=87.0 kgm_1 = 87.0 \, \text{kg}
- Initial velocity of Pablo and his bike, v1=26.0 m/sv_1 = 26.0 \, \text{m/s}
- Mass of the car, m2=1,780 kgm_2 = 1,780 \, \text{kg}
- Initial velocity of the car, v2=0 m/sv_2 = 0 \, \text{m/s} (since the car is stationary)
- Final velocity of Pablo and his bike, v1′=0 m/sv’_1 = 0 \, \text{m/s} (since Pablo comes to a stop)
- Final velocity of the car, v2′v’_2 (the unknown that we need to find)
Step 2: Use the conservation of momentum
The total momentum before the collision is equal to the total momentum after the collision. Initial momentum=Final momentum\text{Initial momentum} = \text{Final momentum}
Before the collision: m1⋅v1+m2⋅v2=(m1+m2)⋅v2′m_1 \cdot v_1 + m_2 \cdot v_2 = (m_1 + m_2) \cdot v’_2
Since v2=0v_2 = 0 and v1′=0v’_1 = 0, the equation simplifies to: m1⋅v1=(m1+m2)⋅v2′m_1 \cdot v_1 = (m_1 + m_2) \cdot v’_2
Step 3: Solve for v2′v’_2
Substitute the known values into the equation: 87.0⋅26.0=(87.0+1,780)⋅v2′87.0 \cdot 26.0 = (87.0 + 1,780) \cdot v’_2 2,262=1,867⋅v2′2,262 = 1,867 \cdot v’_2
Now, solve for v2′v’_2: v2′=2,2621,867≈1.21 m/sv’_2 = \frac{2,262}{1,867} \approx 1.21 \, \text{m/s}
Step 4: Conclusion
The velocity of the car after the collision is approximately 1.21 m/s to the east.
This result is based on the principle of conservation of momentum, which tells us that the total momentum of the system before and after the collision must be equal, assuming no external forces. In this case, since Pablo comes to a stop, the momentum lost by him and his bike is transferred to the car, causing it to move at a velocity of 1.21 m/s.