Question
Kepler's First law (continued) 4. For the ellipse where the semimajor axis 20 AU and the eccentricity =0.5 find a point in the planet's orbit where the values of r_(1) and r_(2) are equal. Sketch the ellipse below as it appears on your screen. (8 points) 5. For the ellipse in Question 4, what is the value of the sum of r_(3) and r_(2) (1 point) __ 6. How does the value in Question 5 mathematically relate to the values of r_(1) and t_(2) (1 point) 7. Play with the simulator, changing the values of the semimajor axis and the eccentricity. Is the relationship you described in Question 6 true for all ellipses? (1 point) Kepler's Second law Complete the steps belo win order. Check them off as you complete them. square Click on the clear optional features button (bottom right) to remove the 1st Law features. square Open the Kepler's 2nd Law tab square Enable the show grid checkbox. square Click on the start sweeping button square Adjust the animation rate so that the planet moves at a reasonable speed. square Experiment with adjusting the size of the sweep using the adjust size slider Click and drag the sweep segment around. Note how the shape of the sweep segment changes, but the area does not
Solution
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Elton
Elite · Tutor for 8 years
Answer
**Kepler's First Law (continued)**4. When r₁ and r₂ are equal, the planet is at one of the two points on the minor axis of the ellipse. These are the points where the ellipse is closest to and furthest from the *center* of the ellipse (not the focus where the sun is). * **Calculating the distances:** We know the semi-major axis (a) is 20 AU and the eccentricity (e) is 0.5. The distance from the center of the ellipse to the focus (c) is given by c = ae = 20 * 0.5 = 10 AU. The semi-minor axis (b) is given by b = a√(1 - e²) = 20√(1 - 0.5²) = 20√(0.75) ≈ 17.32 AU. Therefore, r₁ = r₂ = b ≈ 17.32 AU at the points on the minor axis. * **Sketch:** The sketch should show an ellipse. The Sun should be located at one focus. The major axis is horizontal with a length of 2a = 40 AU. The minor axis is vertical with a length of 2b ≈ 34.64 AU. The points where r₁ and r₂ are equal are at the top and bottom of the minor axis.5. The sum of r₃ (distance from one focus to a point on the ellipse) and r₂ (distance from the other focus to the same point) is equal to 2a (twice the semi-major axis). In this case, 2a = 2 * 20 AU = 40 AU.6. The sum of r₃ and r₂ (40 AU) is equal to the sum of r₁ and t₂ (which should be r₁ and r₂, both of which are approximately 17.32 AU at the points described in question 4). However, this is only true at the specific points on the minor axis. The more general relationship, and the one implied by the question, is that r₃ + r₂ is constant for any point on the ellipse and equal to 2a. The sum of r₁ and r₂ is *not* constant as the planet moves along its orbit. The question is poorly worded and creates confusion between a specific case and the general rule.7. Yes. The relationship r₃ + r₂ = 2a is true for all ellipses. This is a fundamental property of ellipses. The sum of the distances from any point on the ellipse to the two foci is constant and equal to twice the semi-major axis.**Kepler's Second Law**The instructions provided outline the steps to explore Kepler's Second Law using a simulator. Follow those steps within the simulator itself. The key observation is that while the *shape* of the swept area changes as the planet moves along its orbit, the *area* itself remains constant for equal time intervals. This demonstrates that a planet sweeps out equal areas in equal times, which is the essence of Kepler's Second Law.