Question

During a rainy day, rain is falling vertically with a velocity `2m//s` A boy at rest starts his motion with a constant acceleration of `2m//s^(2)` along a straight road . Find the rate at which the angle of the axis of umbrella with vertical should be changed so that the rain always falls parallel to the axis of the umbrella.

Answer 1

Velocity of rain with respect to boy
`vec v_("rain,boy") = vec v_("rain") - vec v_("boy") = vec v_("rain") + (- vec v_("boy"))`
At any time `t`, rain will appear to the boy as shown in (Fig. 5.111).
`tan theta = (at)/(v)`
Boy should hold his umbrella at an angle `theta` from the vertical
`:. tan theta = (at)/(v)`
To find the rate at which the angle of the axis of umbrella with vertical, we need to find `d theta//dt`.
`(d tan theta)/( dt) = d (((at)/(v)))/(d t) rArr sec^2 theta (d theta)/(dt) = (a)/(v)`
`rArr (d theta)/(d t) = (a)/(v sec^2 theta) = (a)/(v [ 1+ tan^2 theta])`
=`(a)/(v[1 + (a^2 t^2)/(v^2)]] = (av)/(v^2 + a^2 t^2)`
=`(2 xx 2)/(4 + 4 t^2) = (1)/(1 + t^2)`
`rArr (d theta)/(d t) = (1)/(1 + t^2)`.
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