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Someone asked me a question on average speed in a different post. So, adding some theory that may help others.
Average speed = \(\frac{Distance}{time}\). We can generally solve such questions by calculating distance over the entire period and the time taken.
For example -
(I) If you are given two speeds s1 for t1 and s2 for t2, the average speed will be \(\frac{s1*t1+s1*t2}{t1+t2}\). This is nothing but \(\frac{Total ..distance}{total..time}\)
(II) If you are given two speeds s1 and s2 over a distance of d1 and d2 respectively, the average speed will be \((d1+d2)/(\frac{d1}{s1}+\frac{d2}{s2})\). This is nothing but \(\frac{Total ..distance}{total..time}\)
The problem comes in when we have to calculate the average speed from just two average speeds with no additional information. Always remember, it will not be the SUM of two speeds over 2 and would require some additional information other than the two below mentioned cases.
Speed of say 10km per hour and 20 km per hour where
1) distance is same say 20 km..
Distance is 20+20=40 and time taken is \(\frac{20}{10}+\frac{20}{20}=2+1=3\).
Average speed is 40/3, which can also be calculated from \(\frac{2*s_1*s_2}{s_1+s_2}=\frac{2*10*20}{10+20}=\frac{400}{30}=\frac{40}{30}.\)
2) time is same say 2 hours
Distance covered = \(s_1*t_1+s_2*t_2=10*2+20*2=60\) km, while time taken is 2+2=4...average speed = 60/4=15, which can also be calculated from \(\frac{s_1+s_2}{2}=\frac{10+20}{2}=15\)
So it is only in these cases that you can find the average speed from two average speed - One when time is same and second when distance is same. In almost all other cases you require some more information - Either time taken or distance traveled over these two speeds.
I will add few OG questions in the thread at a later stage
Average speed = \(\frac{Distance}{time}\). We can generally solve such questions by calculating distance over the entire period and the time taken.
For example -
(I) If you are given two speeds s1 for t1 and s2 for t2, the average speed will be \(\frac{s1*t1+s1*t2}{t1+t2}\). This is nothing but \(\frac{Total ..distance}{total..time}\)
(II) If you are given two speeds s1 and s2 over a distance of d1 and d2 respectively, the average speed will be \((d1+d2)/(\frac{d1}{s1}+\frac{d2}{s2})\). This is nothing but \(\frac{Total ..distance}{total..time}\)
The problem comes in when we have to calculate the average speed from just two average speeds with no additional information. Always remember, it will not be the SUM of two speeds over 2 and would require some additional information other than the two below mentioned cases.
Speed of say 10km per hour and 20 km per hour where
1) distance is same say 20 km..
Distance is 20+20=40 and time taken is \(\frac{20}{10}+\frac{20}{20}=2+1=3\).
Average speed is 40/3, which can also be calculated from \(\frac{2*s_1*s_2}{s_1+s_2}=\frac{2*10*20}{10+20}=\frac{400}{30}=\frac{40}{30}.\)
2) time is same say 2 hours
Distance covered = \(s_1*t_1+s_2*t_2=10*2+20*2=60\) km, while time taken is 2+2=4...average speed = 60/4=15, which can also be calculated from \(\frac{s_1+s_2}{2}=\frac{10+20}{2}=15\)
So it is only in these cases that you can find the average speed from two average speed - One when time is same and second when distance is same. In almost all other cases you require some more information - Either time taken or distance traveled over these two speeds.
I will add few OG questions in the thread at a later stage
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