Question

Question 5⬇

A body covers half of the distance with 15m/s, and the other half is covered in two equal time intervals with velocity 40m/s and 60m/s. Calculate average speed?

Quality answer expected.☺

Answer 1

$\textsf{\Large {\underline {SOLUTION}}}$ :

Let the total distance be x m.

Half of the distance is covered with speed, v = 15 m/s

Distance for the first half part = $\mathsf{\frac{x}{2}}$m.

Time taken to cover first half of the distance = $\mathsf{\frac{x}{2/15}}$ sec.

Time taken, ${t} _{1}$ = $\mathsf{\frac{x}{30}}$ sec.

For other half, the distance is covered in two equal time intervals.

Distance for second half of the journey = $\mathsf{\frac{x}}{2}$m.

${v} _{1}$ = 40 m/s

${v} _{2}$ = 60 m/s

${v} _{1}$ = $\mathsf{\frac{\frac{x}{2}}{t} _{2}}$ = 40 m/s

${v} _{2}$ = $\mathsf{\frac{\frac{x}{2}}{t} _{2}}$ = 60 m/s

Since, time is same. ( t2 )

${t} _{2}$ = $\mathsf{\frac{\frac{x}{2}}{40}}$

${t} _{2}$ = $\mathsf{\frac{\frac{x}{2}}{60}}$

${t} _{2}$ = $\mathsf{\frac{\frac{x}{2}}{40}}$ + $\mathsf{\frac{\frac{x}{2}}{60}}$

${t} _{2}$ = $\mathsf{\frac{x}{80}}$ + $\mathsf{\frac{x}{120}}$

${t} _{2}$ = $\mathsf{\frac{3x}{240}}$ + $\mathsf{\frac{2x}{240}}$

${t}_{2}$ = $\mathsf{\frac{5x}{240}}$

${t}_{2}$ = $\mathsf{\frac{x}{48}}$

Now, Average speed = $\mathsf{\frac{Total\:distance\:covered}{Total\:time\:taken}}$

Average speed, ${v}_{avg}$ = $\mathsf{\frac{x}{t_1+t_2}}$

Average speed, ${v}_{avg}$ = $\mathsf{\frac{x}{\frac{x}{30}\:+\:\frac{x}{48}}}$

Average speed, ${v}_{avg}$ = $\mathsf{\frac{x}{\frac{8x}{240}\:+\:\frac{5x}{240}}}$

Average speed, ${v}_{avg}$ = $\mathsf{\frac{x}{\frac{13x}{240}}}$

Average speed, ${v}_{avg}$ = $\mathsf{\frac{240}{13}}$

➡️ Average speed, ${v}_{avg}$ = 18.46 m/s.