To determine Find the value of v 0 in Equation (5) is the orbit in Equation (6) for a circle, an ellipse, a parabola and a hyperbola. Explanation Formula used: Refer to Equation (5) in the textbook. Write the general expression for Kepler’s first law or ellipse law. e = r 0 v 0 2 G M − 1 (1) Here, v 0 is the initial velocity, r 0 is the initial direction which is a minimum value. G is the universal gravitational constant ( 6.6738 × 10 − 11 Nm 2 kg − 2 ) , and M is the mass of the sun ( 1.99 × 10 30 kg ) . Calculation: Rearrange the equation (1) as follows, r 0 v 0 2 G M = e + 1 v 0 2 = G M ( e + 1 ) r 0 v 0 = G M ( e + 1 ) r 0 (2) For a circle: The eccentricity e is zero. Substitute 0 for e in equation (2) to find v 0 . v 0 = G M ( 0 + 1 ) r 0 = G M r 0 For an ellipse: The eccentricity e is 0 < e < 1 . Substitute 0 for e in equation (2) to find v 0 . v 0 = G M ( 0 + 1 ) r 0 = G M r 0 Substitute 1 for e in equation (2) to find v 0

Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

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Chapter 13.6, Problem 8E

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Find the value of v0 in Equation (5) is the orbit in Equation (6) for a circle, an ellipse, a parabola and a hyperbola.

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Chapter 13.6, Problem 7E

Chapter 13.6, Problem 9E

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Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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