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Chapter 9: Problem 84

Which has a larger magnitude of momentum: a 3000-kg elephant moving at 40 km/h or a 60-kg cheetah moving at \(112 \mathrm{km} / \mathrm{h}\) ?

Short Answer

Expert verified
The 3000-kg elephant moving at 40 km/h has a larger magnitude of momentum (\(33,333.33 kg\cdot\frac{m}{s}\)) than the 60-kg cheetah moving at 112 km/h (\(18,666.67 kg\cdot\frac{m}{s}\)).

Step by step solution

01

Convert speeds from km/h to m/s

Before performing any calculations, it is helpful to have a standard unit of measurement for the velocity. 1 km/h = 1000/3600 m/s For the elephant, we have: \( v_{elephant} = 40\frac{km}{h} * \frac{1000m}{1km} * \frac{1h}{3600s} = \frac{40*1000}{3600} \frac{m}{s} \) For the cheetah, we have: \( v_{cheetah} = 112\frac{km}{h} * \frac{1000m}{1km} * \frac{1h}{3600s} = \frac{112*1000}{3600} \frac{m}{s} \)
02

Calculate the momentum for each animal

Next, we will calculate the momentum of each animal using the formula \( p = mv \), where m is the mass in kg and v is the velocity in m/s. For the elephant: \( p_{elephant} = m_{elephant} * v_{elephant} = 3000 kg * \frac{40*1000}{3600}\frac{m}{s} \) For the cheetah: \( p_{cheetah} = m_{cheetah} * v_{cheetah} = 60 kg * \frac{112*1000}{3600}\frac{m}{s} \)
03

Compare the magnitudes of the momenta

Now that we have expressions for the momenta of both the elephant and the cheetah, we will compare their magnitudes to determine which is larger. For the elephant: \( p_{elephant} = 3000kg * \frac{40*1000}{3600}\frac{m}{s} \) For the cheetah: \( p_{cheetah} = 60kg * \frac{112*1000}{3600}\frac{m}{s} \) Now compare the two momenta: \( 3000kg * \frac{40*1000}{3600}\frac{m}{s} \) vs. \( 60kg * \frac{112*1000}{3600}\frac{m}{s} \) By calculating the values, we get: \( p_{elephant} \approx 33,333.33 kg\cdot\frac{m}{s} \) \( p_{cheetah} \approx 18,666.67 kg\cdot\frac{m}{s} \)
04

Conclusion

Since the momentum of the elephant (\( 33,333.33 kg\cdot\frac{m}{s} \)) is greater than the momentum of the cheetah (\( 18,666.67 kg\cdot\frac{m}{s} \)), the elephant has the larger magnitude of momentum in this scenario.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity Conversion
When dealing with physics problems, it's common to encounter velocities expressed in various units. To simplify calculations, converting these velocities to a standard unit like meters per second (m/s) is essential. In our exercise, both the elephant and the cheetah's speeds were given in kilometers per hour (km/h). To convert km/h to m/s, we use the conversion factor:
  • 1 km/h = 1000 meters / 3600 seconds = 0.27778 m/s.
For the elephant moving at 40 km/h:
  • We set up our conversion by multiplying: \[v_{elephant} = 40 \frac{\text{km}}{\text{h}} \times 0.27778 \frac{\text{m}}{\text{s}} = 11.11 \, \frac{\text{m}}{\text{s}}. \]
For the cheetah at 112 km/h:
  • Similarly, the conversion gives us:\[v_{cheetah} = 112 \frac{\text{km}}{\text{h}} \times 0.27778 \frac{\text{m}}{\text{s}} = 31.11 \, \frac{\text{m}}{\text{s}}. \]
This standardization to m/s makes further calculations straightforward and ensures accuracy.
Momentum Calculation
Momentum (\[ p \]) describes the quantity of motion an object has and is calculated by multiplying the mass of the object by its velocity:
  • \[ p = m \times v \]
Where:
  • \( p \) is momentum (in kg路m/s),
  • \( m \) is mass (in kg), and
  • \( v \) is velocity (in m/s).
For the elephant, which weighs 3000 kg and has a converted velocity of 11.11 m/s:
  • The momentum is: \[p_{elephant} = 3000 \, \text{kg} \times 11.11 \, \frac{\text{m}}{\text{s}} = 33,330 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}. \]
For the cheetah, weighing 60 kg and having a velocity of 31.11 m/s:
  • The momentum is: \[p_{cheetah} = 60 \, \text{kg} \times 31.11 \, \frac{\text{m}}{\text{s}} = 1,866.6 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}. \]
These calculations provide the foundation for comparing the momentum of different moving objects.
Comparison of Momenta
Once we've calculated the momentum for each animal, comparing them is simple. Momentum gives insight into how difficult it would be to stop a moving object. In this exercise:
  • The calculated momentum of the elephant is 33,330 kg路m/s.
  • For the cheetah, the momentum is 1,866.6 kg路m/s.
Comparing these values:
  • The elephant's momentum (33,330 kg路m/s) is significantly larger than the cheetah's (1,866.6 kg路m/s).
This means the elephant, despite being slower, has more momentum because of its much greater mass. The cheetah is fast, but its smaller mass results in less overall momentum. This comparison highlights how both mass and velocity play crucial roles in determining an object's momentum.

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Most popular questions from this chapter

Find the center of mass of a rectangular block of length \(a\) and width \(b\) that has a nonuniform density such that when the rectangle is placed in the \(x, y\) -plane with one corner at the origin and the block placed in the first quadrant with the two edges along the \(x\) - and \(y\) -axes, the density is given by \(\rho(x, y)=\rho_{0} x,\) where \(\rho_{0}\) is a constant.

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Squids have been reported to jump from the ocean and travel \(30.0 \mathrm{m}\) (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of \(20.0^{\circ}\), assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at \(12.0 \mathrm{m} / \mathrm{s} ;\) gravitational force and friction are neglected. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent?

Momentum for a system can be conserved in one direction while not being conserved in another. What is the angle between the directions? Give an example.

What external force is responsible for changing the momentum of a car moving along a horizontal road?
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