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A vehicle travels from city A to city B. During the first part of the journey its average speed is whilst in the rest of the journey it is. By what factor

is the length of the second part of the journey longer than that of the first part if the average speed calculated for

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Asked by satyamsinghtech367 4th July 2022, 9:06 PM
Answered by Expert
Answer:
Part (a)
 
Speed = aritmatic mean of speed of both parts of journey = ( v1 + v2 ) /2
 
Let S1 be the distance travelled in first part and S2 be the distance travelled in second part.
 
We can write the following expression for total time of journey
 
begin mathsize 14px style S subscript 1 over v subscript 1 plus S subscript 2 over v subscript 2 space equals fraction numerator S subscript 1 plus S subscript 2 over denominator left parenthesis v subscript 1 plus v subscript 2 right parenthesis divided by 2 end fraction end style
 
begin mathsize 14px style S subscript 1 over v subscript 1 minus fraction numerator 2 S subscript 1 over denominator left parenthesis v subscript 1 plus v subscript 2 right parenthesis end fraction space equals space fraction numerator 2 S subscript 2 over denominator left parenthesis v subscript 1 plus v subscript 2 right parenthesis end fraction minus S subscript 2 over v subscript 2 end style
 
begin mathsize 14px style S subscript 1 space open parentheses 1 over v subscript 1 minus fraction numerator 2 over denominator left parenthesis v subscript 1 plus v subscript 2 right parenthesis end fraction close parentheses space equals space S subscript 2 open parentheses fraction numerator 2 over denominator left parenthesis v subscript 1 plus v subscript 2 right parenthesis end fraction minus 1 over v subscript 2 close parentheses end style
 
begin mathsize 14px style S subscript 1 over v subscript 1 space cross times fraction numerator stretchy left parenthesis v subscript 2 minus v subscript 1 stretchy right parenthesis over denominator left parenthesis v subscript 1 plus v subscript 2 right parenthesis end fraction space equals space S subscript 2 over v subscript 2 space cross times fraction numerator stretchy left parenthesis v subscript 2 minus v subscript 1 stretchy right parenthesis over denominator left parenthesis v subscript 1 plus v subscript 2 right parenthesis end fraction end style
 
begin mathsize 14px style S subscript 2 over S subscript 1 space equals space v subscript 2 over v subscript 1 end style
--------------------------------------------------------------------------
Part (b)
 
Speed = geometric  mean of speed of both parts of journey = begin mathsize 14px style square root of v subscript 1 v subscript 2 end root end style
 
Let S1 be the distance travelled in first part and S2 be the distance travelled in second part.
 
We can write the following expression for total time of journey
 
begin mathsize 14px style S subscript 1 over v subscript 1 plus S subscript 2 over v subscript 2 space equals fraction numerator S subscript 1 plus S subscript 2 over denominator square root of v subscript 1 v subscript 2 end root end fraction end style
 
begin mathsize 14px style fraction numerator S subscript 1 v subscript 2 space plus space S subscript 2 v subscript 1 over denominator v subscript 1 v subscript 2 end fraction space equals fraction numerator S subscript 1 plus S subscript 2 over denominator square root of v subscript 1 v subscript 2 end root end fraction end style
 
begin mathsize 14px style fraction numerator S subscript 1 v subscript 2 space plus space S subscript 2 v subscript 1 over denominator square root of v subscript 1 v subscript 2 end root end fraction space equals left parenthesis space S subscript 1 plus S subscript 2 space right parenthesis space end style
 
begin mathsize 14px style S subscript 1 v subscript 2 minus space S subscript 1 square root of v subscript 1 v subscript 2 end root space equals space S subscript 2 space square root of v subscript 1 v subscript 2 end root space minus space S subscript 2 space v subscript 1
end style
 
begin mathsize 14px style S subscript 1 square root of v subscript 2 end root space open parentheses square root of v subscript 2 end root space minus space square root of v subscript 1 end root close parentheses space equals space S subscript 2 square root of v subscript 1 end root space open parentheses square root of v subscript 2 end root space minus space square root of v subscript 1 end root close parentheses space end style
 
begin mathsize 14px style S subscript 2 over S subscript 1 space equals space square root of v subscript 2 over v subscript 1 end root end style
---------------------------------------------------------
Part (c)
 
Speed = harmonic  mean of speed of both parts of journey = begin mathsize 14px style fraction numerator 2 space v subscript 1 v subscript 2 over denominator v subscript 1 plus space v subscript 2 end fraction end style
 
Let S1 be the distance travelled in first part and S2 be the distance travelled in second part.
 
We can write the following expression for total time of journey
 
 
begin mathsize 14px style S subscript 1 over v subscript 1 plus S subscript 2 over v subscript 2 space equals open parentheses S subscript 1 plus S subscript 2 close parentheses fraction numerator open parentheses v subscript 1 plus v subscript 2 close parentheses over denominator 2 v subscript 1 v subscript 2 end fraction end style
 
begin mathsize 14px style fraction numerator S subscript 1 v subscript 2 plus S subscript 2 v subscript 1 over denominator v subscript 1 v subscript 2 end fraction space equals open parentheses S subscript 1 plus S subscript 2 close parentheses fraction numerator open parentheses v subscript 1 plus v subscript 2 close parentheses over denominator 2 v subscript 1 v subscript 2 end fraction end style
 
begin mathsize 14px style 2 open parentheses S subscript 1 v subscript 2 space plus space S subscript 2 v subscript 1 close parentheses space equals space open parentheses S subscript 1 plus S subscript 2 close parentheses open parentheses v subscript 1 plus v subscript 2 close parentheses end style
 
begin mathsize 14px style open parentheses S subscript 1 v subscript 2 space plus space S subscript 2 v subscript 1 close parentheses space equals space stretchy left parenthesis S subscript 1 v subscript 1 space plus space S subscript 2 v subscript 2 stretchy right parenthesis end style
 
begin mathsize 14px style S subscript 1 open parentheses v subscript 2 space minus space v subscript 1 close parentheses space equals space S subscript 2 stretchy left parenthesis v subscript 2 space minus space v subscript 1 stretchy right parenthesis end style
 
begin mathsize 14px style S subscript 2 over S subscript 1 space equals space 1 end style
-------------------------------------------------------------------------
Part (d)
 
Speed = weighted arithmatic  mean of speed of both parts of journey = begin mathsize 14px style fraction numerator v subscript 1 plus space k space v subscript 2 over denominator 1 plus space k end fraction end style
 
Let S1 be the distance travelled in first part and S2 be the distance travelled in second part.
 
We can write the following expression for total time of journey
 
 
begin mathsize 14px style S subscript 1 over v subscript 1 plus S subscript 2 over v subscript 2 space equals open parentheses S subscript 1 plus S subscript 2 close parentheses fraction numerator open parentheses 1 space plus k close parentheses over denominator stretchy left parenthesis v subscript 1 plus k space v subscript 2 stretchy right parenthesis end fraction end style
 
 
begin mathsize 14px style S subscript 1 open square brackets 1 over v subscript 1 space minus space fraction numerator 1 plus k over denominator v subscript 1 plus k v subscript 2 end fraction close square brackets space equals space S subscript 2 space open square brackets fraction numerator 1 plus k over denominator v subscript 1 plus k v subscript 2 end fraction space minus space 1 over v subscript 2 space close square brackets end style
 
 
begin mathsize 14px style S subscript 1 space open square brackets fraction numerator k left parenthesis v subscript 2 minus v subscript 1 right parenthesis over denominator v subscript 1 stretchy left parenthesis v subscript 1 plus k space v subscript 2 stretchy right parenthesis end fraction close square brackets space equals space S subscript 2 space open square brackets fraction numerator left parenthesis v subscript 2 minus v subscript 1 right parenthesis over denominator v subscript 2 stretchy left parenthesis v subscript 1 plus k space v subscript 2 stretchy right parenthesis end fraction space close square brackets end style
 
begin mathsize 14px style S subscript 2 over S subscript 1 space equals space k space cross times v subscript 2 over v subscript 1 end style
---------------------------------------------------------------------
Answered by Expert 5th July 2022, 8:28 AM
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