SOLUTION: The average velocity is equal to the average of the initial and final velocities ONLY if the acceleration is constant. In general, average velocity is given by the change in displa

Question 715228: The average velocity is equal to the average of the initial and final velocities ONLY if the acceleration is constant. In general, average velocity is given by the change in displacement divided by the time interval.
Average speed is given by the total distance traveled divided by the total time.
Suppose you travel up a hill at 40 miles/hour and then return down the hill at 60 miles/hour.
a) What was your average speed for the round trip journey?
b) What was your average velocity?
My work for part a:
average speed= (40 mi/hr + 60 mi/hr)/2 = 50 mi/hr
Is my work right? Also, how would part b be solved?
Found 2 solutions by stanbon, DrBeeee:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
The average velocity is equal to the average of the initial and final velocities ONLY if the acceleration is constant.
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Suppose you travel up a hill at 40 miles/hour and then return down the hill at 60 miles/hour.
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Distance up = Distance down = x miles
time up = x/40 mph
time down = x/60 mph
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Average speed is given by the total distance traveled divided by the total time.
a) What was your average speed for the round trip journey?
= total distance/total time = 2x/(x/40 + x/60) = 2x/(100x/2400) = 2*24 = 48 mph
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In general, average velocity is given by the change in displacement divided by the time interval.
b) What was your average velocity?
= 0/(x/40 - x/60) = 0/(20x/2400) = 0
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Cheers,
Stan H.
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Answer by DrBeeee(684) (Show Source): You can put this solution on YOUR website!
Your answer to part a is not quite right because your direction changed at the top of the hill, which means that your acceleration was not constant for the whole trip, up and down. Let's just solve the problem using
(1) d = rt
Let Tup = time taken to go up the hill.
Let Tdn = time to go down the hill.
Let d = the distance up the hill which is the same as the distance down the hill.
Using (1), we get
(2) d = 40*Tup or
(3) Tup = d/40, and
(4) d = 60*Tdn or
(5) Tdn = d/60
The average speed, Save, is given by the total distance traveled divided by the total time taken, giving us
(6) Save = (d + d)/(Tup + Tdn) or
(7) Save = 2d/(d/40 + d/60) or
(8) Save = 2/(1/40 + 1/60) or
(9) Save = 20/(1/4 + 1/6) or
(10) Save = 480/(6 + 4) or
(11) Save = 48
The average speed is 48 miles per hour.
I believe that since we instantaneously reversed directions at the top of the hill the acceleration was infinite for zero seconds, and since the displacement, d, was the same up and down, but in opposite directions the average velocity (vector) is zero. But check with your teacher on the velocity part b.

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