Problem 29 - University Physics with Modern Physics

Question

Changing Mass. An open-topped freight car with mass \(24,000 \mathrm{kg}\) is coasting without friction along a level track. It is raining very hard, and the rain is falling vertically downward. Originally, the car is empty and moving with a speed of 4.00 \(\mathrm{m} / \mathrm{s}\).(a) What is the speed of the car after it has collected 3000 \(\mathrm{kg}\) of rain- water? (b) since the rain is falling downward, how is it able to affect the horizontal motion of the car?

Step-by-Step Solution

Answer

(a) 3.56 m/s. (b) The rain adds mass, leading to decreased speed to conserve momentum.

Step 1: Understanding the Problem

We have an open-topped freight car initially moving with a speed of 4.00 m/s and mass of 24,000 kg. Rainwater falls into the car, adding an additional 3,000 kg mass. We need to find the car's new speed after collecting the rainwater, and explain how vertically falling rain affects the car's horizontal motion.

Step 2: Applying Conservation of Momentum

Since there is no external horizontal force, momentum before and after collecting the rain must be conserved. The initial momentum is given by \( p_i = m_i \times v_i \), where \( m_i = 24,000 \) kg and \( v_i = 4.00 \) m/s. Calculate \( p_i = 24,000 \times 4.00 \).

Step 3: Calculating Initial Momentum

Compute the initial momentum: \( p_i = 24,000 \times 4.00 = 96,000 \) kg·m/s.

Step 4: Setting up Final Momentum Equation

Once the rain has fallen, the total mass \( m_f \) is \( 24,000 + 3,000 = 27,000 \) kg. The final momentum \( p_f \) is \( m_f \times v_f \), where \( v_f \) is the final velocity that we need to find. Set \( p_i = p_f \).

Step 5: Solving for Final Velocity

Equate initial and final momentum: \( 96,000 = 27,000 \times v_f \). Solve for \( v_f \) to find \( v_f = \frac{96,000}{27,000} \).

Step 6: Calculating the Final Speed

Calculate \( v_f = \frac{96,000}{27,000} \approx 3.56 \) m/s. This is the speed of the car after collecting 3,000 kg of rainwater.

Step 7: Explaining the Effect of Vertical Rain

Although rain falls vertically, it adds mass to the system. Since horizontal momentum is conserved and mass increases, the speed must decrease to keep the momentum constant. The rain's vertical motion doesn't impact horizontal momentum directly but affects the result through mass.

Key Concepts

Horizontal Motion

Horizontal motion refers to movement in a straight line parallel to the ground. In physics, when dealing with horizontal motion, we understand that it does not inherently account for vertical forces unless directed by external influence. In our freight car scenario, the car is moving horizontally along a track while rain falls vertically.

You might wonder how a vertical force, like falling rain, impacts this motion. It does not directly affect the direction or movement of the freight car because rain falls vertically with respect to earth's gravity.

• Horizontal momentum remains constant unless acted upon by an external force, according to Newton's First Law.
• Since there's no external horizontal force involved, we say horizontal motion maintains its path and speed, subject to any changes in mass.

So, although the rain changes the mass of the car, it doesn't apply horizontal force to change its direction.

Mass Change

Mass change in a physical system can affect how it moves. In our exercise, rain adds 3,000 kg to the freight car, altering its total mass. When mass changes, especially in a horizontal motion scenario like this, it impacts momentum.

Why does the mass change matter? Because it's directly linked to the momentum of an object moving horizontally. Here's how:

• Initial mass is 24,000 kg and increases by 3,000 kg due to rain.
• New effective mass is now 27,000 kg, which we use to analyze motion changes.

Without the addition of external forces, the system's total momentum before and after must stay the same. Thus, even though the car gains mass, its horizontal speed decreases proportionally to keep momentum conserved.

Velocity Calculation

Velocity calculation is essential to understand how fast something is moving after an interaction like mass change. When considering momentum conservation, we equate the initial and final momentum to solve for velocity.

Here's how it's done using the conservation of momentum:

• Initial momentum of the car is calculated with its starting mass and speed: \( p_i = 24,000 \times 4.00 = 96,000 \; \text{kg} \cdot \text{m/s} \).
• After collecting rain, with the new mass of 27,000 kg, we need to find the new velocity \( v_f \).
• We set \( p_i = p_f \), leading to \( 96,000 = 27,000 \times v_f \).
• Solve for \( v_f \) to find \( v_f = \frac{96,000}{27,000} \approx 3.56 \text{ m/s} \).

This calculated velocity reflects how rainfall changed the system's speed but preserved momentum.