An object moves in such a manner that its velocity at time t is given by
v(t) = t3 - 4t2 +3t
At t = 0, it starts at position x = 0. During the interval [0, 4] what is the absolute distance moved?
you're taking calculus, right?
you have an equation that gives a body's position, and you're asking how far it has moved in a period of time?
duh
(ending position) - (initial position)
To find the absolute distance moved by the object during the interval [0,4], we need to calculate the total displacement over that time period. Displacement is the difference between the initial and final positions.
The velocity function v(t) gives the rate of change of position with respect to time. If we integrate the velocity function over the interval [0,4], we can find the total change in position or displacement.
To find the displacement, we integrate the velocity function as follows:
∫[0,4] (t³ - 4t² + 3t) dt = [¼t⁴ - 4/3t³ + 3/2t²] [0,4]
Evaluating the definite integral, we have:
[¼(4)⁴ - 4/3(4)³ + 3/2(4)²] - [¼(0)⁴ - 4/3(0)³ + 3/2(0)²]
Simplifying:
[¼(256) - 4/3(64) + 3/2(16)] - [0]
Simplifying further:
[64 - 256/3 + 24] - [0]
[64 - 85.33 + 24]
62.67
Therefore, the absolute distance moved by the object during the interval [0,4] is 62.67 units.