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title: "A scientific probe of mass \\(m\\) is launched radially outward from the surface of a spherical planet of radius \\(R\\). For \\(r \\ge R\\), the probe experiences an attractive conservative gravitational force \\(\\vec{F}(r) = -\\dfrac{A}{r^{5/2}}\\hat{r}\\), where \\(A\\) is a positive constant and \\(r\\) is the distance from the center of the planet. The probe is launched with an initial speed \\(v_0 = 2v_{\\text{esc}}\\), where \\(v_{\\text{esc}}\\) is the minimum speed required to escape the planet’s gravitational pull to infinity. Assuming that resistive forces are negligible and the potential energy approaches zero as \\(r \\to \\infty\\), what is the kinetic energy of the probe when it is infinitely far from the planet, in terms of \\(A\\) and \\(R\\)?"
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url: "https://nerd-notes.com/ubq/120664/"
date_modified: "2026-08-23T04:42:32+00:00"
---

# A scientific probe of mass \(m\) is launched radially outward from the surface of a spherical planet of radius \(R\). For \(r \ge R\), the probe experiences an attractive conservative gravitational force \(\vec{F}(r) = -\dfrac{A}{r^{5/2}}\hat{r}\), where \(A\) is a positive constant and \(r\) is the distance from the center of the planet. The probe is launched with an initial speed \(v_0 = 2v_{\text{esc}}\), where \(v_{\text{esc}}\) is the minimum speed required to escape the planet’s gravitational pull to infinity. Assuming that resistive forces are negligible and the potential energy approaches zero as \(r \to \infty\), what is the kinetic energy of the probe when it is infinitely far from the planet, in terms of \(A\) and \(R\)?

A scientific probe of mass \(m\) is launched radially outward from the surface of a spherical planet of radius \(R\). For \(r \ge R\), the probe experiences an attractive conservative gravitational force \(\vec{F}(r) = -\dfrac{A}{r^{5/2}}\hat{r}\), where \(A\) is a positive constant and \(r\) is the distance from the center of the planet. The probe is launched with an initial speed \(v_0 = 2v_{\text{esc}}\), where \(v_{\text{esc}}\) is the minimum speed required to escape the planet's gravitational pull to infinity. Assuming that resistive forces are negligible and the potential energy approaches zero as \(r \to \infty\), what is the kinetic energy of the probe when it is infinitely far from the planet, in terms of \(A\) and \(R\)?

- **A.** \(\dfrac{2A}{3R^{3/2}}\)
- **B.** \(\dfrac{2A}{R^{3/2}}\)
- **C.** \(\dfrac{7A}{3R^{3/2}}\)
- **D.** \(\dfrac{8A}{3R^{3/2}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120664/*