Problem 76 - University Physics with Modern Physics

Question

At a classic auto show, a \(840-\mathrm{kg} 1955\) Nash Metropolitan motors by at 9.0 \(\mathrm{m} / \mathrm{s}\) , followed by a \(1620-\mathrm{kg} 1957\) Packard Clipper purring past at 5.0 \(\mathrm{m} / \mathrm{s}\) . (a) Which car has the greater kinetic energy? What is the ratio of the kinetic energy of the Nash to that of the Packard? (b) Which car has the greater magnitude of momentum? What is the ratio of the magnitude of momentum of the Nash to that of the Packard? (c) Let \(F_{\mathrm{N}}\) be the net force required to stop the Nash in time \(t,\) and let \(F_{\mathrm{P}}\) be the net force required to stop the Packard in the same time. Which is larger: \(F_{\mathrm{N}}\) or \(F_{\mathrm{P}} ?\) What is the ratio \(F_{\mathrm{N}} / F_{\mathrm{P}}\) of these two forces? (d) Now let \(F_{\mathrm{N}}\) be the net force required to stop the Nash in a distance \(d,\) and let \(F_{\mathrm{P}}\) be the net force required to stop the Packard in the same distance. Which is larger: \(F_{\mathrm{N}}\) or \(F_{\mathrm{P}} ?\) What is the ratio \(F_{\mathrm{N}} / F_{\mathrm{P}}\)?

Step-by-Step Solution

Answer

(a) Nash has greater KE, ratio is 1.68. (b) Packard has greater momentum, ratio is 0.93. (c) \( F_P > F_N \), ratio is 0.93. (d) \( F_N > F_P \), ratio is 1.68.

Step 1: Calculate Kinetic Energy for Both Cars

The kinetic energy (KE) of an object is given by the formula \( KE = \frac{1}{2}mv^2 \). - For the Nash, with mass \( m = 840 \, \text{kg} \) and velocity \( v = 9.0 \, \text{m/s} \): \[ KE_{\text{Nash}} = \frac{1}{2} \times 840 \, \text{kg} \times (9.0 \, \text{m/s})^2 = 34020 \, \text{Joules} \]- For the Packard, with mass \( m = 1620 \, \text{kg} \) and velocity \( v = 5.0 \, \text{m/s} \): \[ KE_{\text{Packard}} = \frac{1}{2} \times 1620 \, \text{kg} \times (5.0 \, \text{m/s})^2 = 20250 \, \text{Joules} \]Hence, the Nash has greater kinetic energy.

Step 2: Calculate Kinetic Energy Ratio

To find the ratio of the kinetic energy of the Nash to that of the Packard, use the formula: \[ \text{Ratio}_{KE} = \frac{KE_{\text{Nash}}}{KE_{\text{Packard}}} = \frac{34020}{20250} \approx 1.68 \]

Step 3: Calculate Momentum for Both Cars

The momentum \( p \) of an object is given by \( p = mv \).- For the Nash: \[ p_{\text{Nash}} = 840 \, \text{kg} \times 9.0 \, \text{m/s} = 7560 \, \text{kg} \cdot \text{m/s} \]- For the Packard: \[ p_{\text{Packard}} = 1620 \, \text{kg} \times 5.0 \, \text{m/s} = 8100 \, \text{kg} \cdot \text{m/s} \]Therefore, the Packard has greater momentum.

Step 4: Calculate Momentum Ratio

To find the ratio of the momentum of the Nash to that of the Packard: \[ \text{Ratio}_{p} = \frac{p_{\text{Nash}}}{p_{\text{Packard}}} = \frac{7560}{8100} \approx 0.93 \]

Step 5: Determine Net Force Required to Stop in Time t

The net force required to stop an object in time \( t \) can be found using \( F = \frac{p}{t} \) because \( F \cdot t = p \).- For the Nash: \[ F_{N} = \frac{p_{\text{Nash}}}{t} = \frac{7560}{t} \]- For the Packard: \[ F_{P} = \frac{p_{\text{Packard}}}{t} = \frac{8100}{t} \]Since \( 8100 > 7560 \), \( F_{P} > F_{N} \).

Step 6: Calculate Force Ratio for Stopping in Time t

The ratio of the forces: \[ \text{Ratio}_{F_t} = \frac{F_{N}}{F_{P}} = \frac{7560}{8100} \approx 0.93 \]

Step 7: Determine Net Force Required to Stop in Distance d

When stopping in a distance \( d \), use the work-energy principle: \( F \times d = \Delta KE \). Thus, \( F = \frac{KE}{d} \).- For the Nash: \[ F_{N} = \frac{34020}{d} \]- For the Packard: \[ F_{P} = \frac{20250}{d} \]Since \( 34020 > 20250 \), \( F_{N} > F_{P} \).

Step 8: Calculate Force Ratio for Stopping in Distance d

The force ratio when stopping in distance d: \[ \text{Ratio}_{F_d} = \frac{F_{N}}{F_{P}} = \frac{34020}{20250} \approx 1.68 \]

Key Concepts

Momentum

Momentum is a vital concept in physics that describes the quantity of motion an object has. It is defined as the product of an object's mass and velocity. The formula is simple, yet powerful: \[ p = mv \] where \(p\) is momentum, \(m\) is mass, and \(v\) is velocity.

Key characteristics of momentum include:

• Momentum is a vector quantity, meaning it has both magnitude and direction.
• Objects with greater mass or velocity have higher momentum.
• Momentum plays a significant role in understanding collisions and interactions.

In the auto show scenario, the Packard Clipper, although slower, has more momentum (8100 kg·m/s) than the Nash Metropolitan (7560 kg·m/s) due to its larger mass. This illustrates how both speed and mass influence momentum.

When comparing momentum between objects, it is essential to consider both factors, helping us predict and analyze motion accurately.

Net Force

Net force refers to the total force acting on an object, determining its acceleration according to Newton's second law. This law is expressed as: \[ F = ma \] where \(F\) is net force, \(m\) is mass, and \(a\) is acceleration.

When an object needs to stop, like cars in motion, the net force applied is crucial. The force needed depends on how much momentum the object has and the time or distance over which it is stopped.

• Stopping time: The net forces needed for both the Nash and the Packard to stop in the same time differ due to their momentum. The Packard, with greater momentum, also requires greater force to stop within the same timeframe.
• Stopping distance: If the cars must stop in the same distance, net forces are determined by their kinetic energies. The Nash, with higher kinetic energy, needs more force to halt over that distance.

This concept shows how both time and distance conditions affect stopping forces, highlighting different needs based on an object's physical properties and motion.

Work-Energy Principle

The work-energy principle is a cornerstone in mechanics, connecting the work done on an object to its energy. According to this principle, the work done by forces on an object leads to a change in its kinetic energy.

The formula expressing this is: \[ \Delta KE = F \times d \] where \(\Delta KE\) is the change in kinetic energy, \(F\) is the force applied, and \(d\) is the distance over which the force is applied.

Understanding this principle can be visualized in the scenario where cars need to stop:

• To stop a car, work must be done against its kinetic energy.
• The Nash, with an initial kinetic energy of 34,020 Joules, requires more work (force applied over a distance) to stop than the Packard, which has 20,250 Joules.

As such, for cars with different kinetic energies, the application of force varies to bring them to a halt over the same distance. This principle solidifies how energy interplay affects practical applications like braking.