A small rock passes a massive star, following the path shown in red on the diagram above. When the rock is a distance 4e+13 m (indicated as d1 on the diagram) from the center of the star, the magnitude of its momentum p1 is 1.1e+17 kg ·m/s, and the angle α is 116 degrees. At a later time, when the rock is a distance d2 = 8.4e+12 m from the center of the star, it is heading in the -y direction. There are no other massive objects nearby.
What is the magnitude of the momentum p2?
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To find the magnitude of the momentum p2, we can use the principle of conservation of momentum. Since there are no other massive objects nearby, we know that the momentum of the rock should remain constant.
Therefore, we can say that p1 = p2. So, the magnitude of the momentum at the later time, p2, would be 1.1e+17 kg·m/s.
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To determine the magnitude of the momentum, we can use the principle of conservation of momentum. We know that there are no other massive objects nearby, so the total momentum of the system will remain constant.
Given:
- Distance of the rock from the center of the star when the magnitude of momentum is p1: d1 = 4e+13 m
- Magnitude of the momentum at distance d1: p1 = 1.1e+17 kg·m/s
- Angle of the rock's momentum vector α: α = 116 degrees
- Distance of the rock from the center of the star at a later time: d2 = 8.4e+12 m
Since momentum is a vector quantity, we need to break it down into its x and y components using trigonometry.
Step 1: Determine the x and y components of the momentum at position d1.
Let's assume the x-axis is along the direction of motion of the rock at position d1, and the y-axis is perpendicular to the x-axis.
The x-component of the momentum (p1x) can be calculated using the formula:
p1x = p1 * cos(α)
Substituting the values:
p1x = (1.1e+17 kg·m/s) * cos(116 degrees)
Step 2: Determine the y-component of the momentum at position d1.
The y-component of the momentum (p1y) can be calculated using the formula:
p1y = p1 * sin(α)
Substituting the values:
p1y = (1.1e+17 kg·m/s) * sin(116 degrees)
Step 3: Determine the x and y components of the momentum at position d2.
Since the rock is heading in the -y direction, the x-component of the momentum at position d2 will be 0.
The y-component of the momentum (p2y) at position d2 will be the same as the y-component of the momentum at position d1 since there are no external forces acting on the system to change the y-component of the momentum.
Step 4: Determine the magnitude of the momentum at position d2.
The magnitude of the momentum (p2) can be calculated using the Pythagorean theorem:
p2 = sqrt(p2x^2 + p2y^2)
Since p2x = 0 (as determined in Step 3): p2 = sqrt(0 + p2y^2) = p2y
Therefore, to find the magnitude of the momentum at position d2, we need to determine the y-component of the momentum at position d1.
Note that without the values for p1x and p1y, we cannot calculate the magnitude of the momentum p2 accurately. The provided information does not give the necessary data to answer the question.
To determine the magnitude of the momentum (p2) of the rock when it is at distance d2 from the center of the star, we can use the principle of conservation of momentum.
The conservation of momentum states that in the absence of external forces, the total momentum of a system remains constant. In this case, since there are no other massive objects nearby to exert external forces on the rock, its momentum will be conserved.
To solve for p2, we can use the following fundamental equation:
p1 = p2 [Conservation of momentum]
Given that the magnitude of momentum at the initial position (d1) is 1.1e+17 kg ·m/s (p1), we can set up the equation as follows:
1.1e+17 kg ·m/s = |p2| [Note that we discard the negative sign for magnitude]
Since we only need the magnitude of p2, we can disregard the direction and focus on the absolute value.
Therefore, the magnitude of the momentum (p2) of the rock at distance d2 from the center of the star would also be 1.1e+17 kg ·m/s.