A mouse travels along a straight line; its distance from the origin at any time is given by the equation.

x= (8.1 cm * s^-1)t-(2.7cm * s^-2)t^2

A) Find the average velocity of the mouse in the interval from t= 0 to t= 1.0 .
value=?; unit= ?

B)Find the average velocity of the mouse in the interval from t=0 to t= 4.0 .
value?; unit= ?

Express your answer to two significant figures and include the appropriate units.

A) To find the average velocity from t = 0 to t = 1.0, we need to calculate the change in position (x) divided by the change in time (t).

At t = 0, the position of the mouse is x = (8.1 cm/s)(0) - (2.7 cm/s^2)(0)^2 = 0 cm.

At t = 1.0, the position of the mouse is x = (8.1 cm/s)(1.0 s) - (2.7 cm/s^2)(1.0 s)^2 = 5.4 cm.

The change in position is 5.4 cm - 0 cm = 5.4 cm, and the change in time is 1.0 s - 0 s = 1.0 s.

So, the average velocity is 5.4 cm / 1.0 s = 5.4 cm/s.

B) To find the average velocity from t = 0 to t = 4.0, we can follow the same process as above.

At t = 0, the position of the mouse is x = (8.1 cm/s)(0) - (2.7 cm/s^2)(0)^2 = 0 cm.

At t = 4.0, the position of the mouse is x = (8.1 cm/s)(4.0 s) - (2.7 cm/s^2)(4.0 s)^2 = -70.8 cm.

The change in position is -70.8 cm - 0 cm = -70.8 cm, and the change in time is 4.0 s - 0 s = 4.0 s.

So, the average velocity is -70.8 cm / 4.0 s = -17.70 cm/s.

Therefore, the average velocity of the mouse in the interval from t = 0 to t = 1.0 is 5.4 cm/s, and from t = 0 to t = 4.0 is -17.70 cm/s.

To find the average velocity of the mouse, we need to calculate the displacement and divide it by the time interval.

A) Average velocity from t = 0 to t = 1.0:

To determine the displacement, we need to find the position of the mouse at t = 0 and t = 1.0.

At t = 0:
x(0) = (8.1 cm/s * (s^-1))(0) - (2.7 cm/s^2 * (s^-2))(0)^2
x(0) = 0

At t = 1.0:
x(1.0) = (8.1 cm/s * (s^-1))(1.0) - (2.7 cm/s^2 * (s^-2))(1.0)^2
x(1.0) = 8.1 cm/s - 2.7 cm/s^2 * (1.0)^2
x(1.0) = 8.1 cm/s - 2.7 cm/s^2 * 1.0
x(1.0) = 5.4 cm/s

The displacement is the difference between the final and initial positions:
Displacement = x(1.0) - x(0)
Displacement = 5.4 cm/s - 0
Displacement = 5.4 cm/s

The average velocity is determined by dividing the displacement by the time interval:
Average Velocity = Displacement / Time Interval
Average Velocity = 5.4 cm/s / (1.0 - 0)
Average Velocity = 5.4 cm/s

Therefore, the average velocity of the mouse in the interval from t = 0 to t = 1.0 is 5.4 cm/s.

B) Average velocity from t = 0 to t = 4.0:

We will follow the same steps as above to find the displacement.

At t = 4.0:
x(4.0) = (8.1 cm/s * (s^-1))(4.0) - (2.7 cm/s^2 * (s^-2))(4.0)^2
x(4.0) = 8.1 cm/s * 4.0 - 2.7 cm/s^2 * 16.0
x(4.0) = 32.4 cm/s - 43.2 cm/s
x(4.0) = -10.8 cm/s

Displacement = x(4.0) - x(0)
Displacement = -10.8 cm/s - 0
Displacement = -10.8 cm/s

Average Velocity = Displacement / Time Interval
Average Velocity = -10.8 cm/s / (4.0 - 0)
Average Velocity = -2.7 cm/s

Therefore, the average velocity of the mouse in the interval from t = 0 to t = 4.0 is -2.7 cm/s.

To find the average velocity of the mouse in each time interval, we need to calculate the displacement and divide it by the time interval. The displacement can be found by evaluating the position equation at the given values of t.

A) Average velocity from t = 0 to t = 1.0:
To find the displacement, we substitute t = 1.0 into the position equation:
x = (8.1 cm/s)(1.0 s) - (2.7 cm/s^2)(1.0 s)^2
x = 8.1 cm - 2.7 cm
x = 5.4 cm

The displacement is 5.4 cm over a time interval of 1.0 second, so the average velocity is:
Average velocity = displacement / time interval = 5.4 cm / 1.0 s = 5.4 cm/s

Therefore, the average velocity of the mouse in the interval from t = 0 to t = 1.0 is 5.4 cm/s.

B) Average velocity from t = 0 to t = 4.0:
Using the same process, we substitute t = 4.0 into the position equation:
x = (8.1 cm/s)(4.0 s) - (2.7 cm/s^2)(4.0 s)^2
x = 32.4 cm - 43.2 cm
x = -10.8 cm

The displacement is -10.8 cm over a time interval of 4.0 seconds, so the average velocity is:
Average velocity = displacement / time interval = -10.8 cm / 4.0 s = -2.7 cm/s

Therefore, the average velocity of the mouse in the interval from t = 0 to t = 4.0 is -2.7 cm/s.

A) Divide the distance traveled by the time interval

[x(1) - x(0)]/1 = (8 - 2.7)/1 = 5.3 m/s

B) Do the same thing, but use t(4), t(0) and an interval of t = 4.
Note that the mouse changes direction and ends up going backwards.

[x(4) - x(0)]/4 = (32.4 - 43.2]/4
= -2.7 m/s