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Average velocity, as I have heard, cannot be found simply by finding the average of two numbers. I have a question on average velocity, but am simply unable to proceed:

A particle moving in a straight line covers half the distance with speed of 3m/s. The other half of the distance is covered in two equal time intervals with speed 4.5m/s and 7.5m/s respectively. The average speed of the particle during this motion is?

What I know: The average speed is the total distance divided by the total time How exactly should I approach the question?

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  • $\begingroup$ Have you tried finding the total distance and dividing by the total time? $\endgroup$ – Jordan Jun 13 '14 at 18:59
  • $\begingroup$ Can't find the distance :( $\endgroup$ – Gummy bears Jun 13 '14 at 19:00
  • $\begingroup$ you don't have to assume it any arbitrary variable $\endgroup$ – DSinghvi Jun 13 '14 at 19:05
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v1=3, v2=4.5 , v3=7.5

let total distance be s

so average speed =total distance/total time.

total distance =s

time for 1st half distance = s/2/v1

for second half distance covered = s/2 =distance covered in 2 equal time intervals (say time interval be t)=t*4.5+t*7.5=t(4.5+7.5)=t(12)

so here t={s/2}/12

average speed={s}/[{s/2}/3+s/24+s/24]=s/{6s/24}=4m/s since 2t was time for second journey

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  • $\begingroup$ I think that's wrong, the answer is supposed to be 4 m/s $\endgroup$ – Gummy bears Jun 13 '14 at 19:08
  • $\begingroup$ "total distance =s time for 1st half distance = s/v1" seems like typo, you need to divide s by 2. $\endgroup$ – DavePhD Jun 13 '14 at 19:10
  • $\begingroup$ you are right @Gummybears $\endgroup$ – DSinghvi Jun 13 '14 at 19:16
  • $\begingroup$ Yeah, but that's cause I have the answer key. I don't get how to reach that answer. Using his method, I too reached the same answer :( $\endgroup$ – Gummy bears Jun 13 '14 at 19:18
  • $\begingroup$ I have edited see.@Gummybears $\endgroup$ – DSinghvi Jun 13 '14 at 19:19
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Hint: Pick a distance. It doesn't matter, it will divide out. $45$ meters is handy. Now calculat the total time required from the information given. We know the first $22.5$ m was covered at $3$ m/sec, so how many seconds were required? How many seconds are required for the other two pieces? Now divide your distance by the total time.

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You can average the speeds in the second half, since they are for equal time intervals.

Let the total distance be a variable (d).

Solve for the time for each half segment, in terms of d.

Add the times together to get the total time.

Apply the definition you mention: "average speed is the total distance divided by the total time".

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  • $\begingroup$ But for the second half, they are divided in terms of time! Please help! $\endgroup$ – Gummy bears Jun 13 '14 at 19:10
  • $\begingroup$ I edited to explain more. $\endgroup$ – DavePhD Jun 13 '14 at 19:19

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