The position of a particle on the x-axis at time t, t>0' is ln t. Find the average velocity of the particle for 1<=t<=e .
(e-1)intergal ln t dt from 1 to e
(e-1) [ 1/e - 1/1]
sorry, all you need is position at t = e - position at t = 1 divided by (e-1)
[ ln e - ln 1/ / (e-1) = 1/(e-1)
To find the average velocity of the particle for the given time interval, we need to calculate the displacement and divide it by the time taken.
Displacement is given by the difference in position between the final and initial times:
Δx = x(final) - x(initial)
Let's calculate the displacement:
For t = 1, the position of the particle is ln(1) = 0.
For t = e, the position of the particle is ln(e) = 1.
Therefore, the displacement is:
Δx = 1 - 0 = 1
Now, let's calculate the time taken:
The given time interval is 1 ≤ t ≤ e, where e is a constant approximately equal to 2.71828.
Therefore, the time taken is:
Δt = e - 1
Now, we can calculate the average velocity:
Average velocity = displacement / time taken
Average velocity = Δx / Δt
Substituting the values:
Average velocity = 1 / (e - 1)
Hence, the average velocity of the particle for 1 ≤ t ≤ e is 1 / (e - 1).
To find the average velocity of the particle, we need to calculate the displacement of the particle over the given time interval and divide it by the time elapsed.
The position of the particle on the x-axis at time t is given as ln t. The displacement of the particle can be calculated by subtracting the initial position from the final position.
Let's find the position of the particle at the beginning (t = 1) and at the end (t = e) of the given time interval.
For t = 1, the position of the particle is ln 1 = 0, since the natural logarithm of 1 is 0.
For t = e, the position of the particle is ln e = 1, since the natural logarithm of e is 1.
So, the displacement of the particle is 1 - 0 = 1.
The time elapsed is e - 1.
To find the average velocity, we divide the displacement by the time elapsed:
Average velocity = displacement / time elapsed
= 1 / (e - 1)
Therefore, the average velocity of the particle for 1 <= t <= e is 1 / (e - 1).