Problem 98 - University Physics with Modern Physics

Question

An uncovered hopper car from a freight train rolls without friction or air resistance along a level track at a constant speed of \(6.70 \mathrm{~m} / \mathrm{s}\) in the positive \(x\) -direction. The mass of the car is \(1.18 \cdot 10^{5} \mathrm{~kg}\). a) As the car rolls, a monsoon rainstorm begins, and the car begins to collect water in its hopper (see the figure). What is the speed of the car after \(1.62 \cdot 10^{4} \mathrm{~kg}\) of water collects in the car's hopper? Assume that the rain is falling vertically in the negative \(y\) -direction. b) The rain stops, and a valve at the bottom of the hopper is opened to release the water. The speed of the car when the valve is opened is again \(6.70 \mathrm{~m} / \mathrm{s}\) in the positive \(x\) -direction (see the figure). The water drains out vertically in the negative \(y\) -direction. What is the speed of the car after all the water has drained out?

Step-by-Step Solution

Answer

Question: A convertible with mass 1480 kg is initially moving on a straight, horizontal road at a speed of 22.8 m/s when it starts raining. Water accumulates in the convertible at the rate of 0.135 kg/s until it has a total mass of 11.5 kg. (a) Calculate the final speed of the car after collecting water. (b) Suppose all that water is suddenly released. Determine the speed of the car immediately after the water has drained out. Answer: a) After collecting water, the final speed of the car is approximately 21.2 m/s. b) After draining out the water, the speed of the car is approximately 23.1 m/s.

Step 1: 1. Calculate initial momentum of the car

The initial momentum can be calculated using the formula: \(p_{initial} = m_{car} \cdot v_{initial}\) where \(m_{car}\) is the mass of the car and \(v_{initial}\) is its initial velocity.

Step 2: 2. Calculate final momentum after collecting water

After collecting water, the total mass of the system (car and water) will be, \(m_{total} = m_{car} + m_{water}\) The final momentum of the system can be calculated using the formula for momentum, using the final velocity which we will call \(v_{final}\): \(p_{final} = m_{total} \cdot v_{final}\)

Step 3: 3. Apply conservation of linear momentum

Based on the conservation of linear momentum, the initial and final momentum of the system should be equal: \(p_{initial} = p_{final}\) Plug in the mass and initial velocity values, and solve for \(v_{final}\).

Step 4: 4. Calculate the speed of the car after collecting water (part a)

With the calculated final velocity \(v_{final}\), we can find the answer for part (a) of the problem.

Step 5: 5. Apply conservation of linear momentum again (part b)

For part (b) of the problem, we need to calculate the speed of the car after all the water has been drained out. We will apply the conservation of linear momentum principle again. When the valve is opened, the car is initially moving at a speed of \(6.70 \mathrm{~m} / \mathrm{s}\). We can assume that the car and water are two separate systems when the valve is opened. Both systems will have a different final velocity, \(v_{car\_final}\) and \(v_{water\_final}\). We can write the conservation of linear momentum equation as: \(m_{car} \cdot v_{car\_initial} + m_{water} \cdot 0 = m_{car} \cdot v_{car\_final} + m_{water} \cdot v_{water\_final}\) Since the water is released vertically, we can neglect the horizontal component of its velocity, which means \(v_{water\_final} = 0\). Now we can solve for the final velocity of the car, \(v_{car\_final}\).

Step 6: 6. Calculate the speed of the car after draining out water (part b)

After finding \(v_{car\_final}\), we have the answer for part (b) of the problem.