a point moving on a straight line traversed half the distance with a velocity v0. the remaining part of the distance was covered with velocity v1 for half the time, and with velocity v2 for the other half of the time. find the mean velocity of the point averaged over the whole time.

Question

a point moving on a straight line traversed half the distance with a velocity v0. the remaining part of the distance was covered with velocity v1 for half the time, and with velocity v2 for the other half of the time. find the mean velocity of the point averaged over the whole time.

Answer 1

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Answer 2

To find the mean velocity of the point averaged over the whole time, we need to consider the total distance traveled and the total time taken.

Let's denote the total distance as D and the total time as T. We know that the point traveled half the distance (D/2) with velocity v0. This means that it took (D/2)/v0 = D/(2v0) time to cover this distance.

The remaining distance is also D/2. For this distance, the point traveled with velocity v1 for half the time and velocity v2 for the other half of the time.

Let's calculate the time taken for each velocity. The time taken for velocity v1 is (D/2)/(2v1) = D/(4v1). Similarly, the time taken for velocity v2 is (D/2)/(2v2) = D/(4v2).

So the total time, T, is the sum of the times taken for each velocity:
T = D/(2v0) + D/(4v1) + D/(4v2)

Now, let's calculate the total distance traveled. The point traveled D/2 distance with velocity v0 and D/2 distance with either v1 or v2. So the total distance, D, is given by:
D = D/2 + D/2

Simplifying this equation, we get D = D.

Now, let's substitute the value of D into the expression for T:
T = D/(2v0) + D/(4v1) + D/(4v2)

Since D is common in all three terms, we can factor it out and simplify further:
T = D * (1/(2v0) + 1/(4v1) + 1/(4v2))

Finally, the mean velocity, V_avg, is defined as the total distance divided by the total time:
V_avg = D/T

Substituting the expression for D and T, we have:
V_avg = (D) / (D * (1/(2v0) + 1/(4v1) + 1/(4v2)))
V_avg = 1 / (1/(2v0) + 1/(4v1) + 1/(4v2))

Therefore, the mean velocity of the point averaged over the whole time is given by the expression:
V_avg = 1 / (1/(2v0) + 1/(4v1) + 1/(4v2))

Answer 3

do + d1 +d2 = d

do = d/2
d1+d2 = d/2

to + t1 + t2 = t
to = do/Vo = d/2Vo

t2 = t1
t1 = d1/V1
t2 = d2/V2
so
d1/V1 =d2/V2
d2 = d1 V2/V1
so
t = d/2Vo + d1/V1 + d2/V2
average speed = d/t
= d / (d/2Vo + d1/V1 + d2/V2)
or
d/(d/2Vo + 2 d1/V1)