A car travels at a speed of 20 km/h for half the distance and at 30 km/h for the rest as shown in the diagram.
Find the average speed .
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Assume the distance is 120 km, then the total time taken is 3 + 2 = 5 hours => the average is 120/5 = 24 km/h
Proof that this method works for any x:
Time taken over first half: x/20 h
Time taken over second half: x/30 h
Total time: x/20 + x/30 = 5x/60 h
Total average time: total distance / total time = 2x / (5x/60) = 120/5 = 24 km/h for ANY x
So you might as well take a convenient value for x, like 60 as I did.
The average speed is v = t i m e d i s t a n c e . The time needed to take half the distance at 20km/h is 2 0 k m / h d i s t a n c e / 2 and for the other half of the trip 3 0 k m / h d i s t a n c e / 2 . The total time is t i m e = 2 0 k m / h d i s t a n c e / 2 + 2 0 k m / h d i s t a n c e / 2 . Using this we get
v = t i m e d i s t a n c e = 2 0 k m / h d i s t a n c e / 2 + 2 0 k m / h d i s t a n c e / 2 d i s t a n c e = 2 0 1 + 2 0 1 2 = 2 4 k m / h
Let the total distance be D . The time taken to travel the first half of the distance, t 1 = 2 0 2 D = 4 0 D h and that for the second half t 2 = 3 0 2 D = 6 0 D h . The average speed v a v = total distance traveled ÷ total time taken. ⟹ v a v = t 1 + t 2 D = 4 0 D + 6 0 D D = 5 1 2 0 = 2 4 km/h .
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I have 2 ways of solving this problem:
(1) Using physics
Get this diagram:
• In this t 1 represents time taken from starting point to the center whereas t 2 represents time taken from center to the finishing point. Moreover x determines the distance travelled.
Average speed = Total time Total distance = t 1 + t 2 2 x − − − − − − − − − − − − − > This is equation 1
• Since it is uniform velocity over each sections of x , therefore we can use s = u t because velocity is constant for x but not for 2 x .
⟹ s = u t x = 2 0 × t 1 − − − − − − − − − − − − − > This is equation 2 x = 3 0 × t 2 − − − − − − − − − − − − − > This is equation 3
• Now substitute equation 2 and 3 in equation 1 as follows:
Average speed = t 1 + t 2 2 x = 2 0 x + 3 0 x 2 x = x ( 2 0 1 + 3 0 1 ) 2 x ⟹ ( 2 0 1 + 3 0 1 ) = 24 km/h 2
(2)Using mathematics
Directly use the Harmonic mean formula:
f HM = a 1 1 + ⋯ + a k 1 k .
Or else in our case:
⟹ 2 0 1 + 3 0 1 = 24 km/h 2
Some common errors are:
You try to assume a distance. This is incorrect because time will change according to the distance.
You try to use the arithmetic mean which will give you 25 km/h. It may be close but not accurate enough.