1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# The differential equation of a body fired vertically from the earth is given by,vdvdt=−gr2x2.The initial velocity of a body supposed to escape is?

A
3gr
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
2gr
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
gr
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
5gr
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

## The correct option is B √2grWe have vdvdx=−gr2x2.This is a differential equation of variable separable type.∴vdv=gr2⋅dxx2By intergration, v22=gr2x+cIf u is the required velocity on the surface of the earth where x=r thenu22=gr+c ∴c=u22−gr∴v22=gr2x+u22−gr∴v2=2gr2x+u2−2grThis is the equation of motion of a body projected from the surface of the earth with initial velocity u.If the body is not to return to the earth its velocity v must be always positive. (If the velocity v becomes zero the body will to rest and then will start to descend.) As x increases. 2gr2/x decreases. Hence, v will be positive itu2−2gr≥0 ∴u2≥2gr i.e u≥√2gr.∴ The least velocity of projection =√2gr A particle projected with this velocity will never return to the earth. This is called the escape velocity from the earth.

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos