Question

The speed of a boat is given by, v = k(^1⁄t – at), where k is the constant and l us the distance travel by boat in time t and a is the acceleration of water. If there is an change in ‘l’ from 2cm to 1cm in time 2sec to 1sec. If the acceleration of water changes from 0.95 mt/sec^2 to 2 mt/sec^2 find the motion of boat.

(a) -2

(b) 2

(c) 0.5

(d) -0.5

The question was posed to me in quiz.

Query is from Errors and Approximations topic in portion Partial Differentiation of Engineering Mathematics

Answer 1

Correct option is (a) -2

For explanation I would say: Given, v = k(^1⁄t – at)

Differentiating it we get

\(δv=k\left [\frac{(tδl-lδt)}{t^2} – aδt – tδa \right ]\)

Hence,

\(\frac{δv}{v}=\frac{k\left [\frac{(tδl-lδt)}{t^2} – aδt – tδa \right ]}{v}\)

\(\frac{δv}{v}=\frac{\left [\frac{(tδl-lδt)}{t^2} – aδt – tδa \right ]}{(\frac{l}{t}-at)}\)

Put, l = 2cm, t = 2sec, a = 0.95 mt/sec^2

and  δl = 1cm, δt = 1 sec⁡and δa = -1.05 mt/sec^2

we get,

^δv⁄v = 2.