Question

A ball thrown horizontal from a height speed v travel a horizontal distance R from dimensional analysis find a possible dependence of R on H, v and g​

Answer 1

Answer:

The horizontal distance R covered by the body thrown with a horizontal velocity V, from a tower of height H will be equal to the horizontal velocity V and the time it would the body to fall through height H if just dropped from top of the tower.

Now the body is dropped from height H will fall to ground in time to. Time to can be determined from the equation,

s = u t + ½ g t², in our problem s= H, u =0 m/s,

so H = ½ g t², therefore

t = (2H/ g )½ and therefore R = V× ( 2H/g)½

Let us try to do it using dimensional analysis.

R = H^a V^b g^c

Now dimensions of

R = L,

V= L T^-1

From dimensional analysis

L = L^a L^b T^-b L^c T^-2= L^a+b+c T^-(b+2c)

Comparing powers of L, The on two sides we get

a + b +c = 1,

b + 2 c = 0. Or b =- 2 c or a - 2c + c =1, or a- c=1

or a+ b+c = 1

a - c = 1

Since there are two equations and three unknown we cannot solve unless we assume one value. Let us take, b= 1. This will give c= -½ and a = ½ and R becomes,

R= H½ × V¹ × g-½ = V(H/g)½ .

Dimensional analysis has its limitations. It does not give multiplicative dimensionless constants. It fails like here when unknown are more than equations.