How fast to double your average speed?

Question

The beach is x kilometers away . You drove to the beach at a rate of 60km/hr. How fast would you have to drive back to have gone the whole round trip at an average speed of 120km/hr?

I'm posting this in reference to another question I answered as a demonstration.

Answer 1

Since the average velocity is defined by displacement/time, then

V_avg = (2x)/(x/60 + x/v)

V_avg = 2/(1/60 + 1/v)

120 = 2/(1/60 + 1/v)

60 = 1/(1/60 + 1/v)

1/60 = 1/60 + 1/v

0 = 1/v

It cannot be done.

This is because for the average velocity to be 120, you need to be home by the time you've just made it to the beach. So there is no finite velocity to solve the problem.

Of course, this is applying the physics definition of "average velocity." In maths, where the original question was asked, you have to also be told, "Average with respect to what variable?" If it's an average with respect to position instead of time, the other responses in the thread are correct. The original question is ill-posed. (surpise!) :P

Answer 2

This is an easy one.
x km at 60 km/hr gives y = x/60 hr (time taken)
and
total distance ÷ total time = 120
so ( x + w) / (y +z) = 120 and we know y in terms of x
so (x+w) / (x/60 + z) = 120 or x+w = 2x + 120z
or w = x + 120z
and since the question is what is w/z we replace w with x + 120 z
so w/z = (x+120z)/z = x/z +120
example
lets say x = 60 km so y = 1hr
giving 120 = (60+w)/(1+z) lets say z is 2 hr
the speed w/z is 60/2 + 120 = 150
check (60 + 300)/(1+2) = 360/3 = 120 yup.
just make sure that at the end of two hours at 150 kph you find yourself "back". So, I guess the answer is any arbitrary speed greater than 120 for a time which will be greater as the speed approaches 120.
Gee, you weren't foolishly assuming that the distance traveled to get back would be the same thereby requiring infinite speed? Thanks for the fair warning.

Answer 3

We assume that the whole round trip takes into account the firs x km trip at 60 km/hr (otherwise the problem would be trivial, just drive at 120 km/hr any distance back and forth).

We assume also that you do not change the direction of your travel, for you say "drive back".

We need to travel at a speed v > 120 in order to reach the 120 average. We have

120 = (x + 2 x') / (total time) = (x + 2 x') / (x / 60 + 2 x' /v)

so we obtain the formula

(v-120) / v = x / (2 x')

relating v with x'. If we choose for instance v = 240, then we get

x' = x.

If we choose v = 160 km/h then we need x ' = 2 x.

Te conclusion is that our answer will depend on the new speed value.

Answer 4

Average anything is the sum of the parts (S) divided by the number of parts (N). In your case S = v0 + v1; where v0 = 60 kph going to the beach and v1 = velocity returning home. The number of parts N = 2 because you have two velocities (actually speeds, the magnitudes of the velocities, because you do not give direction).

Thus, by definition, average velocity (speed) V = S/N = (v0 + v1)/N = 120 kph = (60 + v1)/2; so that 240 = 60 + v1 and v1 = 180 kph, which is way too fast on those winding roads.

You should note the distinction between speed and velocity. Speed is just the magnitude of a velocity, it has no implied direction. Velocity is both magnitude (speed) and direction. If v0 = 60 kph due east and v1 = 180 kph due west, then S = 60 E - 180 W = 120 W; that is, the vector sum of the two velocities is a net velocity going 120 kph west. In which case, the average V, as a vector, would be S/N = 120 kph W/2 = 60 kph west velocity, where it was 120 kph as a speed.

Answer 5

you're mendacity, with the aid of fact that's impossible. the only thank you to double your generic speed is to transport abode from the sea coast. enable's say the sea coast is ten miles away. Then the area there and returned is 20 miles. You drove at 10 mph, so your holiday there took an hour. you at the instant choose your generic speed to be 20 mph, meaning you will possibly force 20 miles in an hour. yet 20 miles is the completed distance, and you have already pushed for an hour. except you will get abode in 0 time, there is not any way for this to artwork.