at time t, the position of a body moving along the s-axis is s=t^3-12t^2+36t m. find the total distance traveled by the body from t=0 to t=3
find s when t=3
find s when t = 1
the difference between these two s values is the distance traveled
To find the total distance traveled by the body from t = 0 to t = 3, we need to calculate the displacement at each time interval and add up the absolute values of the displacements.
The displacement at each time interval can be calculated by taking the difference between the position at the end of the interval and the position at the start of the interval.
Given that the position of the body at time t is s = t^3 - 12t^2 + 36t, let's calculate the displacement at each interval.
When t = 0, the position is:
s(0) = (0)^3 - 12(0)^2 + 36(0)
s(0) = 0
When t = 3, the position is:
s(3) = (3)^3 - 12(3)^2 + 36(3)
s(3) = 27 - 108 + 108
s(3) = 27
The displacement from t = 0 to t = 3 is:
displacement = s(3) - s(0)
displacement = 27 - 0
displacement = 27 m
Since we are interested in the total distance traveled, we take the absolute value of the displacement to get:
total distance = |displacement|
total distance = |27|
total distance = 27 m
Therefore, the total distance traveled by the body from t = 0 to t = 3 is 27 meters.
To find the total distance traveled by the body from t=0 to t=3, we need to calculate the distance for each interval where the velocity changes direction.
1. Find the velocity function by taking the derivative of the position function with respect to time.
Given: s(t) = t^3 - 12t^2 + 36t
Velocity function: v(t) = s'(t) = 3t^2 - 24t + 36
2. Find the points where the velocity is equal to zero by solving the equation v(t) = 0.
3t^2 - 24t + 36 = 0
Use the quadratic formula to solve for t. The equation has two roots: t1 and t2.
3. Calculate the distance traveled for each interval by taking the absolute value of the difference between the position values at corresponding points.
Distance1 = |s(t1) - s(0)|
Distance2 = |s(t2) - s(t1)|
Distance3 = |s(3) - s(t2)|
4. Compute the total distance traveled by summing up the distances from each interval.
Total distance traveled = Distance1 + Distance2 + Distance3
By following these steps, you should be able to find the total distance traveled by the body from t=0 to t=3.