11.1 Solved examples

  Motion in One Dimension

Question: Rain drops are falling vertically with respect to ground with a speed of $4\ m/s$. If a boy starts running with a speed of $3\ m/s$ in horizontal direction. Find the direction in which he should hold the umbrella.

Solution: Writing all the information as below,

${\overrightarrow v _R} = - 4\widehat j$
${\overrightarrow v _B} = 3\widehat i$

From the concept of relative motion we can write,
$${\overrightarrow v _{RB}} = {\overrightarrow v _R} - {\overrightarrow v _B}$$$${\overrightarrow v _{RB}} = - 4\widehat j - 3\widehat i$$or$${\overrightarrow v _{RB}} = - 3\widehat i - 4\widehat j$$
The man holds the umbrella in the direction of ${\overrightarrow v _{RB}}$.

Therefore,
$$\tan \theta = \frac{3}{4}$$$$\theta = 37^\circ $$
The boy will hold the umbrella making angle of $37^\circ$ with the vertical.

Question: A man is coming down on an incline of angle $30^\circ$. When he walks with a speed of $2\sqrt 3 \,m/s$ he keeps his umbrella vertical to protect himself from the rain. The actual speed of the rain is $5\ m/s$. Find the velocity of the rain.

Solution:

$\left| {{{\overrightarrow v }_R}} \right| = 5\,m/s$
${\overrightarrow v _M} = 2\sqrt 3 \left( {\cos 30^\circ \widehat i - \sin 30^\circ \widehat j} \right)$
${\overrightarrow v _{RM}} = a\widehat j$ (As he keeps his umbrella vertical).

Let the velocity of the rain be,
$${\overrightarrow v _R} = b\widehat i + c\widehat j$$Also, $$\sqrt {{b^2} + {c^2}} = 5 \quad ...(i)$$
From the concept of relative motion we can write,
$${\overrightarrow v _{RM}} = {\overrightarrow v _R} - {\overrightarrow v _M}$$$$a\widehat j = \left( {b\widehat i + c\widehat j} \right) - \left[ {2\sqrt 3 \left( {\cos 30^\circ \widehat i + \sin 30^\circ \widehat j} \right)} \right]$$$$a\widehat j = \left( {b - 3} \right)\widehat i + \left( {c + \sqrt 3 } \right)\widehat j$$So, $$b = 3\quad ...(ii)$$ From equation $(i)$ and $(ii)$ we get,
$$\sqrt {{3^2} + {c^2}} = 5$$$$9 + {c^2} = 25$$$${c^2} = 16$$$$c = \pm 4$$
So, the velocity of the rain is,
$${\overrightarrow v _R} = 3\widehat i \pm 4\widehat j$$