The position of a particle moving along the x axis is given by x = (21 + 22t − 6t^2)m, where t is in s. What is the average velocity during the time interval t = 1s to t = 3s?

Well, well, well... time to calculate some average velocity, huh? Don't worry, I'm here to make it fun!

To find the average velocity, we need to calculate the displacement of the particle and divide it by the time interval. So let's get started.

First, we find the position of the particle at t = 1s:
x(1) = 21 + 22(1) - 6(1)^2
x(1) = 37m

Next, we find the position of the particle at t = 3s:
x(3) = 21 + 22(3) - 6(3)^2
x(3) = 3m

Now let's calculate the displacement:
Displacement = x(3) - x(1)
Displacement = 3m - 37m
Displacement = -34m

So, the particle has a displacement of -34m during the time interval from t = 1s to t = 3s.

Finally, let's calculate the average velocity:
Average velocity = Displacement / Time interval
Average velocity = -34m / (3s - 1s)
Average velocity = -34m / 2s
Average velocity = -17m/s

And there you have it! The average velocity during the time interval from t = 1s to t = 3s is -17m/s. Keep those particles moving, but remember to watch your x-axis speed limit!

To find the average velocity during the time interval from t = 1s to t = 3s, we need to calculate the total displacement of the particle during that time period and divide it by the time elapsed.

1. Start by finding the position of the particle at t = 1s:
Plug t = 1 into the equation x = (21 + 22t − 6t^2)m:
x = 21 + 22(1) − 6(1)^2
x = 21 + 22 − 6
x = 37m

2. Next, find the position of the particle at t = 3s:
Plug t = 3 into the equation x = (21 + 22t − 6t^2)m:
x = 21 + 22(3) − 6(3)^2
x = 21 + 66 − 54
x = 33m

3. Compute the total displacement during the time interval:
Displacement = x(final) - x(initial)
Displacement = 33m - 37m
Displacement = -4m

4. Determine the elapsed time:
Elapsed time = t(final) - t(initial)
Elapsed time = 3s - 1s
Elapsed time = 2s

5. Calculate the average velocity:
Average velocity = Displacement / Elapsed time
Average velocity = -4m / 2s
Average velocity = -2m/s

Therefore, the average velocity during the time interval from t = 1s to t = 3s is -2 m/s.

To find the average velocity of a particle during a given time interval, we can use the formula:

Average Velocity = (Change in position) / (Change in time)

In this case, we need to find the change in position and the change in time.

Given that the position of the particle is given by the equation x = (21 + 22t − 6t^2)m, we can substitute the values of t = 1s and t = 3s into the equation to find the positions at those times.

For t = 1s:
x = (21 + 22(1) − 6(1)^2)m
x = (21 + 22 − 6)m
x = 37m

For t = 3s:
x = (21 + 22(3) − 6(3)^2)m
x = (21 + 66 − 54)m
x = 33m

Now, we can find the change in position (Δx) by subtracting the initial position from the final position:
Δx = x2 - x1
Δx = 33m - 37m
Δx = -4m

The change in time (Δt) is simply the difference between the final and initial times:
Δt = t2 - t1
Δt = 3s - 1s
Δt = 2s

Now, we can use the formula for average velocity:

Average Velocity = (Change in position) / (Change in time)
Average Velocity = Δx / Δt
Average Velocity = -4m / 2s
Average Velocity = -2 m/s

Therefore, the average velocity of the particle during the time interval t = 1s to t = 3s is -2 m/s.

distance/time

at t = 3, x = 21 + 22(3) -6(9)
at t = 1, x = 21 + 22(1) -6(1)
subtract
divide by 2 seconds