To determine the speed of the truck after 50 seconds, we need to apply the principle of conservation of momentum. Since the truck is moving on a frictionless surface, there are no external forces acting on it, and the momentum of the system must remain constant.
Initial Conditions
Let's break down the problem:
- Initial velocity of the truck, u = 10 m/s
- Initial mass of the truck, m_truck = 100 kg
- Initial mass of water, m_water_initial = 100 kg
- Rate of water leaking, r = 2 kg/s
- Time duration, t = 50 s
Calculating the Mass of Water After 50 Seconds
First, we need to find out how much water leaks out in 50 seconds:
m_water_leaked = r × t = 2 kg/s × 50 s = 100 kg
Since the initial mass of the water was also 100 kg, the remaining mass of water after 50 seconds is:
m_water_final = m_water_initial - m_water_leaked = 100 kg - 100 kg = 0 kg
Total Mass of the Truck After 50 Seconds
Now, the total mass of the truck after 50 seconds is simply the mass of the truck since all the water has leaked out:
m_total_final = m_truck + m_water_final = 100 kg + 0 kg = 100 kg
Applying Conservation of Momentum
According to the conservation of momentum, the initial momentum of the system must equal the final momentum:
Initial Momentum = Initial Mass × Initial Velocity
P_initial = (m_truck + m_water_initial) × u = (100 kg + 100 kg) × 10 m/s = 2000 kg·m/s
Final Momentum = Final Mass × Final Velocity
P_final = m_total_final × v_final = 100 kg × v_final
Setting Up the Equation
Setting the initial momentum equal to the final momentum gives us:
2000 kg·m/s = 100 kg × v_final
Solving for Final Velocity
Now, we can solve for v_final:
v_final = 2000 kg·m/s / 100 kg = 20 m/s
Conclusion
After 50 seconds, the speed of the truck is 20 m/s. Since this value is not listed in the options provided (1) 12 m/s (2) 10 m/s (3) 5 m/s (4) None of these, the correct answer is (4) None of these.
