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Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique. Concept Questions Object A is moving due east, while object B is moving due north. They collide and stick together in a completely inelastic collision. Momentum is conserved. (a) Is it possible that the two-object system has a final total momentum of zero after the collision? (b) Roughly, what is the direction of the final total momentum of the two-object system after the collision? Problem Object A has a mass of \(m_{A}=17.0 \mathrm{~kg}\) and an initial velocity of \(\overrightarrow{\mathbf{v}}_{0 \mathrm{~A}}=8.00 \mathrm{~m} / \mathrm{s},\) due east. Object \(\mathrm{B},\) however, has a mass of \(m_{\mathrm{B}}=29.0 \mathrm{~kg}\) and \(\mathrm{an}\) initial velocity of \(\overrightarrow{\mathbf{v}}_{0 \mathrm{~B}}=5.00 \mathrm{~m} / \mathrm{s},\) due north. Find the magnitude and direction of the total momentum of the two-object system after the collision. Make sure that your answers are consistent with your answers to the Concept Questions.

Step by step solution

01

Analyze the Concept Questions (a)

In a completely inelastic collision, momentum is conserved. After the collision, the total momentum of the system equals the sum of the initial momenta of objects A and B. Since both objects start with non-zero momentum in perpendicular directions, it is not possible for their total final momentum to be zero unless they have equal magnitude in opposite directions, which isn't the case here.

02

Analyze the Concept Questions (b)

The final momentum direction can be found by adding the vector components of the initial momenta. Since object A is moving due east and object B is moving due north, the resulting direction will typically lie in the northeast quadrant, specifically, it will form an angle to the east of north, dependent on the magnitudes of the initial momenta.

03

Calculate the Initial Momentum in the East Direction

Calculate the initial momentum of object A: \[ p_{A} = m_{A} imes v_{0A} = 17.0 \, \text{kg} \times 8.00 \, \text{m/s} = 136 \, \text{kg m/s} \] This is in the eastward direction.

04

Calculate the Initial Momentum in the North Direction

Calculate the initial momentum of object B: \[ p_{B} = m_{B} imes v_{0B} = 29.0 \, \text{kg} \times 5.00 \, \text{m/s} = 145 \, \text{kg m/s} \] This is in the northward direction.

05

Determine the Final Momentum Components

Since the collision is completely inelastic and momentum is conserved, the final momentum components simply equal the sum of the initial momentum components. East component = 136 kg m/s, North component = 145 kg m/s.

06

Calculate the Magnitude of the Total Final Momentum

Use the Pythagorean theorem to calculate the magnitude of the total final momentum:\[ p_{\text{final}} = \sqrt{(136^2 + 145^2)} = \sqrt{(18496 + 21025)} = \sqrt{39521} = 198.8 \, \text{kg m/s} \]

07

Determine the Direction of the Final Momentum

Use trigonometry to determine the angle of the final momentum vector with respect to the east direction:\[ \theta = \tan^{-1}\left(\frac{145}{136}\right) \] Calculating gives \( \theta \approx 46.2^\circ \). Thus, the final momentum direction is \(46.2^\circ\) north of east.

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Key Concepts

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

Momentum Conservation

In physics, momentum conservation is a fundamental concept. It's the principle that within a closed system, the total momentum remains constant if no external forces are acting on it. This principle is vital in understanding collisions, such as the one described in the problem where two objects collide and stick together in an inelastic collision.

For the exercise, momentum is given by the formula:

• \( p = m \times v \)
  where:
  • \( p \) is momentum,
  • \( m \) is mass, and
  • \( v \) is velocity.

Even after the objects stick together, the system's overall momentum remains the same as before the collision when calculated as the vector sum of both momenta.

Vector Components

When dealing with two-dimensional motion, it's essential to break down vectors into their components. Each vector has both magnitude and direction, which can be split into horizontal and vertical components. In our scenario, the horizontal and vertical components are along the east and north directions, respectively.

To find these components:

• Determine the initial momentum of each object individually.
• Use their respective directions to assign components:
  • For the east component: Object A's momentum is entirely eastward.
  • For the north component: Object B's momentum is entirely northward.

This breakdown allows us to analyze how each object's motion contributes to the total motion after they collide.

Momentum Direction

The direction of the momentum vector after a collision is crucial to determine as it helps understand the resultant movement of the combined mass. For this problem, the direction of the final momentum after the collision will be a combination of the components found earlier.

To find the precise direction, we do the following:

• Add the vector components found from the initial momenta of both objects.
• The final direction is given by the angle formed by the resultant vector with respect to one of the axes, usually calculated using the inverse tangent function.

The resultant direction shows the path in which the combined object will continue to move post-collision.

Trigonometry in Physics

Trigonometry plays a vital role in physics, especially in problems dealing with angles and vector components. In this collision problem, trigonometry helps calculate the exact angle that the momentum vector makes relative to a reference direction.

Typically, this involves using inverse trigonometric functions:

• \( \theta = \tan^{-1}\left(\frac{\text{north component}}{\text{east component}}\right) \)
  This formula computes the angle \( \theta \) that the momentum vector makes with the east direction.

By understanding and applying these trigonometric principles, you can determine how momentum's direction relates to the standard compass directions. This application of trigonometry is invaluable in resolving real-world physics problems that involve directions and angles.