Question

Solve the following problem.

A train is moving eastward at 10 m/sec. A waiter is walking eastward at 1.2m/sec; and a fly is charging toward the north across the waiter’s tray at 2 m/s. What is the velocity of the fly relative to Earth?

Answer 1

Given:
velocity of train w.r.t Earth, ${\displaystyle {\overrightarrow{\text{v}}}_{\text{TE}}=10\hat{\text{i}}}$
velocity of waiter w.r.t train, ${\displaystyle {\overrightarrow{\text{v}}}_{\text{WT}}=1.2\hat{\text{i}}}$ 
velocity of fly w.r.t waiter, ${\displaystyle {\overrightarrow{\text{v}}}_{\text{FW}}=2\hat{\text{j}}}$

∴ Velocity of fly with respect to Earth

${\displaystyle {\overrightarrow{\text{v}}}_{\text{FE}}={\overrightarrow{\text{v}}}_{\text{FT}}-{\overrightarrow{\text{v}}}_{\text{ET}}}$

${\displaystyle =({\overrightarrow{\text{v}}}_{\text{FW}}-{\overrightarrow{\text{v}}}_{\text{TW}})-{\overrightarrow{\text{v}}}_{\text{ET}}}$

${\displaystyle =2\hat{\text{j}}-(-1.2\hat{\text{i}})-(-10\hat{\text{i}})}$

${\displaystyle =2\hat{\text{j}}+11.2\hat{\text{i}}}$   .....(considering north along +y axis)

Magnitude = ${\displaystyle \sqrt{{11.2}^2+2^2}}$

= 11.38 m/s ≈ 11.4 m/s

Direction of velocity,

${\displaystyle \theta ={\tan }^{-1}\left(\frac{2}{11.2}\right)\approx {10}^{\circ }}$ towards north of east.