Question

A particle balls towards Earth from infinity its velocity on reaching the earth will be our is radius of earth

Answer 1

According to me , your question is -------> find velocity of particle when particle falls towards earth from infinity ?

We know, Total energy = kinetic energy + potential energy [ when external energy is neglected ]
kinetic energy of particle at earth surface = 1/2mv²
Here, m is mass of particle and v is velocity of particle at Earth's surface.

potential energy of particle = -GMm/r
Here, M is mass of earth and r is the radius of earth.
Totol energy of particle at infinity = 0

∴ 0 = 1/2mv² - GMm/r
⇒1/2 mv² = GMm/r
⇒v² = 2GM/r
⇒v = √{2GM/r}
Here, G = 6.67 × 10⁻¹¹ Nm²/kg²
M = 6 × 10²⁴ kg
r = 6400 km = 6.4 × 10⁶ m
∴ v = √{2 × 6.67 × 10⁻¹¹ × 6 × 10²⁴/6.4 × 10⁶}
= √{2 × 6.67 × 6 × 10⁷/6.4}
= √{2 × 66.7 × 6/6.4} × 10³ m/s
≈ 11 km/s

Hence, velocity of particle at earth surface is 11 km/s