The escape velocity v of a body depends on– (i) the acceleration du

1.	The escape velocity v of a body depends on– (i) the acceleration due to gravity ‘g’ of the planet, (ii) the radius R of the planet. Establish dimensionally the relation for the escape velocity.
Answer» v α g^a R^b ⇒ v = k g^a R^b, K → dimensionless proportionality constant [v] = [g]^a [R]^b [M⁰L¹T^–1] = [M⁰L¹T^–2]^a [M⁰ L¹ T⁰]^b equating powers 1 = a + b – 1 = – 2a ⇒ a = 1/2 b = 1 – a = 1 - 1/2 = 1/2 ∴ v = k √gR

The escape velocity v of a body depends on– (i) the acceleration due to gravity ‘g’ of the planet, (ii) the radius R of the planet. Establish dimensionally the relation for the escape velocity.

Answer»

v α g^a R^b ⇒ v = k g^a R^b,

K → dimensionless proportionality constant

[v] = [g]^a [R]^b

[M⁰L¹T^–1] = [M⁰L¹T^–2]^a [M⁰ L¹ T⁰]^b

equating powers 1 = a + b

– 1 = – 2a ⇒ a = 1/2

b = 1 – a = 1 - 1/2 = 1/2

∴ v = k √gR