Mike leaves the house on a bike at 9am. Amy leaves 3 hours later, and goes 21km/h faster than Mike. If Amy overtakes Mike after 196km, what is average speed of both vehicles?
after 3 hours, at speed m, Mike has gone 3m km ahead.
The speed difference is 21 km/hr, so it will take Amy 3m/21 = m/7 hours to overtake Mike.
So, m/7 * (m+21) = 196
m = 28
So, the average speed is (28+(28+21))/2 = 38.5 km/hr
On the other hand, the weighted average (totaldistance/totaltime) is (7*28+4*49)/(7+4) = 35.6 km/hr
since more time was spent at the slower speed.
Thanks Steve, how do you know to divide your 3m by the difference in speed?
The difference is speed determines how long it will take for Amy to catch up. The rest of her speed is the same as Mike's, so she does not gain any distance except for the extra speed she has. Play with some simple numbers and you can see that this is so.
Ok great thanks!
To find the average speed of both vehicles, we need to determine the time it takes for Amy to overtake Mike and the total distance covered.
Let's start by finding the time it takes for Amy to catch up with Mike. Since Amy leaves 3 hours later, we can say that both Mike and Amy have been traveling for the same amount of time before Amy overtakes Mike. Let's represent this time as 't.'
Since Mike left at 9 am, and Amy left 3 hours later, Amy caught up with Mike at 9 am + t hours. The time it took for Amy to catch up is also the time they traveled before Amy caught up with Mike.
Now, we need to determine the distance covered by both Mike and Amy before they meet. We know that Amy overtakes Mike after 196 km. So, the distance traveled by Amy is 196 km.
Since Amy's speed is 21 km/h faster than Mike's speed, we can express their speeds as follows:
Amy's speed = Mike's speed + 21 km/h
Now we can set up an equation using the formula distance = speed × time for each person:
For Mike: Distance covered by Mike = Mike's speed × t
For Amy: Distance covered by Amy = Amy's speed × t
Since Amy overtakes Mike after 196 km, we can equate the distances covered by each person:
Amy's distance = Mike's distance + 196 km
Amy's speed × t = Mike's speed × t + 196 km
Now we can substitute Amy's speed using the earlier expression:
(Mike's speed + 21 km/h) × t = Mike's speed × t + 196 km
Let's simplify the equation:
Mike's speed × t + 21 km/h × t = Mike's speed × t + 196 km
21 km/h × t = 196 km
Now we need to solve for t, the time it takes for Amy to overtake Mike:
t = 196 km / 21 km/h
t ≈ 9.33 hours
So, it takes approximately 9.33 hours for Amy to overtake Mike.
Now, let's find the total distance covered by both Mike and Amy:
Total distance = Amy's distance = 196 km
Finally, we can find the average speed of both vehicles:
Average speed = Total distance / Total time
First, we need to find the total time:
Total time = Mike's time + Amy's time
Total time = t + 3 hours (since Amy left 3 hours later)
Total time = 9.33 hours + 3 hours
Total time ≈ 12.33 hours
Now we can calculate the average speed:
Average speed = 196 km / 12.33 hours
Average speed ≈ 15.9 km/h
Therefore, the average speed of both vehicles is approximately 15.9 km/h.