A few years ago, the spacecraft Pioneer 10 and 11 were passingthrough the Kuiper

Question

A few years ago, the spacecraft Pioneer 10 and 11 were passingthrough the Kuiper belt, a region beyond Neptune that containsdust, ice, and other small bodies.

a) The spacecraft with mass m, velocity v, and a dish antenna ofradius R were observed to be slowing down with an acceleration -a.Assume that the observed negative acceleration was caused byperfectly inelastic collisions between the spacecraft's antenna andKuiper belt particles that were initially at rest. If the densityof such particles was kg/m^3, find in terms of thespacecraft parameters (m, v, R).

b) Pioneer 10 has a mass of 240 kg, was traveling with a velocityof 12.2 km/s, and carries a dish antenna with a radius R = 1.73 m.The observed acceleration was a = 8E-10 m/s^2. It is known thatsome gaseous nebulae have densities of about 10E-16 kg/cm^3. Is thepioneer acceleration consistent with this number?

Explanation / Answer

F = d P / dt   where F is the external force actingon the system   (F = 0) F = d (M v) / dt = v dM / dt + M dv / dt dv / dt = - (1 / M) v dM / dt M = V = A x M / t = A v dv / dt = - A v2 / M = a a = 1 * 10E-10 * 1.732 * (1.22 *10E4)2 / 240 = 5.83 * 10E-4m/s2 This is consistent only if the density were 10E-16kg/m3 in which case a = 5.83 * 10E-10 m/s2 in which case a = 5.83 * 10E-10 m/s2

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