Question

A particle is fired vertically upward with a speed of 15 km $s^{-1}$ With what speed will it move in intersteller space. Assume only earth's gravitational field.

Answer 1

Initial velocity of the particle = 15 km/s Let its speed be V at intersteller space
$\therefore \left(\frac{1}{2}\right)m[15\times {10}^3-v^2]={\int }_R^{\infty }\frac{GMm}{x^2}dx\Rightarrow \left(\frac{1}{2}\right)m\left[\right(15\times {10}^3{)}^2-V^2]=GMm\left[\frac{-1}{x}\right]\Rightarrow \left(\frac{1}{2}\right)m[(225\times {10}^5)-v^2]=\frac{GMm}{R}\Rightarrow 225\times {10}^5-v^2\frac{2\times 6.67\times {10}^{-11}\times 6\times {10}^{24}}{6400\times {10}^3}\Rightarrow v^2=225\times {10}^6-\frac{40.02}{32}\times {10}^8=2.25\times {10}^8-1.2\times {10}^8={10}^8(1.05)orv=1.01\times {10}^{4}m/s=10km/s$