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?

Short answer

Answer: The final speed of the car after collecting water is approximately 5.97 m/s, and the final speed after all the water drains out is approximately 7.43 m/s.

Step-by-step solution

Part (a): Find the final speed of the car after collecting water.

We will use the conservation of linear momentum to solve this part. The initial and final momentum of the car and water system must be equal as there is no external force acting in the x-direction. Initial momentum of the car: \(p_{car\,initial} = m_{car}v_{initial}\), where \(m_{car} = 1.18 \times 10^5 \, kg\) and \(v_{initial} = 6.70 \, m/s\). Final momentum of car and water: \(p_{car+water\,final} = (m_{car} + m_{water})v_{final}\), where \(m_{water}= 1.62 \cdot 10^4 \, kg\). Using the conservation of momentum, we get: \(p_{car\,initial} = p_{car+water\,final}\), or \(m_{car}v_{initial}=(m_{car}+m_{water})v_{final}\). Now, solve for \(v_{final}\).

Part (a): Calculation of the final speed.

We can solve the above equation for the final speed of the car \(v_{final}\): \(v_{final} = \frac{m_{car}v_{initial}}{m_{car} + m_{water}}\). Substituting the known values, we get: \(v_{final} = \frac{(1.18 \times 10^5\, kg)(6.70\, m/s)}{(1.18\, ×10^5\, kg) + (1.62\, ×10^4\, kg)}\) \(v_{final} \approx 5.97 \, m/s\) The final speed of the car after collecting water is approximately \(5.97 \, m/s\).

Part (b): Find the final speed of the car after the water drains out.

For the second part of the question, we will again use the conservation of linear momentum. The momentum of the car and water system must be conserved after the complete drainage of water. The car's momentum will only change due to the water draining from the valve. Initial momentum of the car and water system: \(p_{car+water\,initial} = (m_{car} + m_{water})v_{initial\,car+water}\), where \(v_{initial\,car+water} = 6.70 \, m/s\). Final momentum of the car: \(p_{car\,final} = m_{car}v_{final\,car}\). Using the conservation of momentum, \(p_{car+water\,initial} = p_{car\,final}\). Now, solve for \(v_{final\,car}\).

Part (b): Calculation of the final speed after the water drains out.

We can solve the above equation for the final speed of the car \(v_{final\,car}\): \(v_{final\,car} = \frac{(m_{car} + m_{water})v_{initial\,car+water}}{m_{car}}\). Substituting the known values, we get: \(v_{final\,car} = \frac{((1.18 \times 10^5\, kg) + (1.62 \times 10^4\, kg))(6.70\, m/s)}{(1.18\, ×10^5\, kg)}\) \(v_{final\,car} \approx 7.43 \, m/s\) The final speed of the car after all the water has drained out is approximately \(7.43 \, m/s\).

Key concepts

Linear Momentum

Linear momentum is a fundamental concept in physics that relates to the motion of objects. It is a vector quantity, meaning it has both a magnitude and a direction. Mathematically, linear momentum (\textbf{p}) is the product of an object's mass (\textbf{m}) and its velocity (\textbf{v}) and is given by the equation:
\[ p = mv \]
When considering the exercise provided, we can see that the hopper car has a significant mass and velocity, granting it a considerable amount of linear momentum. This value is crucial because in a closed system, where no external forces are acting, linear momentum is conserved. Understanding this allows us to predict the motion of objects after interactions, such as the collection or release of rainwater in the exercise scenario.

Linear momentum is not just about how fast something is moving; it's about how much 'oomph' it has when moving. A large, slow-moving object can have the same momentum as a small, fast-moving one if their mass and velocity products are equal. This concept is pivotal in solving problems involving collisions and interactions in mechanical systems.

Momentum Conservation in Physics

The law of conservation of momentum is a key principle in mechanics and underlines how the total momentum of an isolated system remains constant if it is not affected by external forces. According to this principle, the momentum before and after an event must be the same.

This conservation law can be formulated as: \[ p_{initial} = p_{final} \]
Applied to our exercise, when the hopper car begins to fill with water, the momentum of the system (car plus the collected water) remains constant. Provided there is no external force in the x-direction, like friction or air resistance, the car slows down as the mass of the system increases, ensuring momentum conservation. In contrast, when water is drained from the hopper, the car speeds up because its mass decreases, yet the total momentum of the system is still conserved throughout the process.

Remember, conservation laws like this one are the 'accounting rules' of physics; they keep track of quantities that must remain stable in the absence of external disturbances. When approaching a new problem, identifying if any conservation laws apply can often provide a quick and reliable path to the solution.

Mechanical Systems

A mechanical system typically refers to any system of bodies that interact through forces and can include anything from simple levers to complex machines like engines or robots. Understanding how different components of a mechanical system interact is critical in predicting the system's behavior.

In our exercise scenario, the hopper car and water can be seen as a simple mechanical system. The key to analyzing such a system is to look at how the motion of its parts is related, often through conservation laws or equations of motion. For the car, as it collects and releases water, we're observing how its internal state (mass distribution) changes and how those changes affect its motion via the conservation of linear momentum.

Mechanical systems are vital in a wide range of fields from engineering to biology, and concepts like momentum play a significant role in understanding and designing these systems. Elegant and powerful, they use simple principles to explain complex phenomena, ensuring everything from bridges to vehicles operates safely and predictably.