To solve this problem, we need to apply the principle of conservation of momentum. When the two trucks meet and exchange sacks of rice, the total momentum of the system before the exchange must equal the total momentum after the exchange, assuming no external forces act on the system. Let's break this down step by step.
Initial Momentum Calculation
Before the exchange, we can calculate the total momentum of the system. The momentum of an object is given by the formula:
Momentum (p) = mass (m) × velocity (v)
For Truck 1 (mass M, speed v1):
p1_initial = M × v1 = 200 kg × 50 m/s = 10000 kg·m/s
For Truck 2 (mass M, speed v2):
p2_initial = M × v2 = 200 kg × 200 m/s = 40000 kg·m/s
Since the trucks are moving towards each other, we consider the direction of Truck 2's momentum as negative:
p2_initial = -40000 kg·m/s
Now, the total initial momentum (p_initial) of the system is:
p_initial = p1_initial + p2_initial = 10000 kg·m/s - 40000 kg·m/s = -30000 kg·m/s
Momentum After the Exchange
After the exchange of sacks, the masses of the trucks become:
- Truck 1: M + m = 200 kg + 50 kg = 250 kg
- Truck 2: M + m = 200 kg + 50 kg = 250 kg
Let’s denote the new velocities of Truck 1 and Truck 2 after the exchange as v1' and v2', respectively. The total momentum after the exchange can be expressed as:
p_final = (M + m) × v1' + (M + m) × (-v2')
Substituting the masses:
p_final = 250 kg × v1' - 250 kg × v2'
Applying Conservation of Momentum
According to the conservation of momentum:
p_initial = p_final
Substituting the values we calculated:
-30000 kg·m/s = 250 kg × v1' - 250 kg × v2'
Setting Up the Equations
We can simplify this equation:
-30000 = 250(v1' - v2')
Dividing both sides by 250 gives:
v1' - v2' = -120 m/s
This means:
v1' = v2' - 120
Finding the New Velocities
Now we need another equation to solve for v1' and v2'. We can use the fact that the relative velocities of the trucks before the exchange will be equal to the relative velocities after the exchange, but in opposite directions. Thus, we can set up the equation:
v1 + v2 = v1' + v2'
Substituting the known values:
50 + 200 = v1' + v2'
250 = v1' + v2'
Solving the System of Equations
Now we have a system of two equations:
- v1' - v2' = -120
- v1' + v2' = 250
We can solve these equations simultaneously. From the first equation, we can express v1' in terms of v2':
v1' = v2' - 120
Substituting this into the second equation:
(v2' - 120) + v2' = 250
2v2' - 120 = 250
2v2' = 370
v2' = 185 m/s
Now substituting back to find v1':
v1' = 185 - 120 = 65 m/s
Final Results
After the exchange of sacks, the new velocities of the trucks are:
- Truck 1 (v1'): 65 m/s
- Truck 2 (v2'): 185 m/s
This analysis shows how momentum is conserved in a system, even when objects interact with each other. The exchange of mass (the sacks of rice) alters the velocities of the trucks, but the total momentum remains constant throughout the process.
