In the time interval from 0.0 s to 10.1 s, the acceleration of a particle traveling in a straight line is given by ax = (0.1 m/s3)t. Let to the right be the +x direction. The particle initially has a velocity to the right of 10.0 m/s and is located 5.4 m to the left of the origin.
(a) Determine the velocity as a function of time during the interval. (Use the following as necessary: t.)
(b) Determine the position as a function of time during the interval. (Use the following as necessary: t.)
(c) Determine the average velocity between t = 0.0 s and 10.1 s.
To find the velocity as a function of time during the interval, we can use the equation of motion:
v = v0 + at,
where:
v = final velocity,
v0 = initial velocity,
a = acceleration, and
t = time.
Given the initial velocity (v0) as 10.0 m/s, and the acceleration (a) as (0.1 m/s^3)t, we can substitute these values into the equation:
v = 10.0 m/s + (0.1 m/s^3)t.
Therefore, the velocity as a function of time during the interval is v = 10.0 + 0.1t m/s.
To find the position as a function of time during the interval, we can integrate the velocity equation with respect to time.
Using the integration rule:
∫ (0.1t) dt = (0.1/2) t^2,
we get:
x = x0 + ∫ [10.0 + (0.1t)] dt
= x0 + ∫ 10.0 dt + ∫ (0.1t) dt
= x0 + 10.0t + (0.1/2) t^2,
where:
x = final position,
x0 = initial position, and
t = time.
Given the initial position (x0) as -5.4 m, we can substitute these values into the equation:
x = -5.4 m + 10.0t + (0.1/2) t^2.
Therefore, the position as a function of time during the interval is x = -5.4 + 10.0t + (0.1/2) t^2 m.
To calculate the average velocity between t = 0.0 s and 10.1 s, we can use the formula:
Average velocity = (change in position) / (change in time).
Given that the initial time is 0.0 s and the final time is 10.1 s, and using the position equation x = -5.4 + 10.0t + (0.1/2) t^2 m, we can substitute these values to find the change in position:
Change in position = x(10.1 s) - x(0.0 s).
Substituting t = 10.1 s, we get:
Change in position = [-5.4 + 10.0(10.1) + (0.1/2) (10.1)^2] - [-5.4 + 10.0(0.0) + (0.1/2) (0.0)^2].
Simplify this expression to find the change in position.
Then, calculate the change in time: t(final) - t(initial).
Finally, divide the change in position by the change in time to calculate the average velocity between t = 0.0 s and 10.1 s.