Questions: A bee with a velocity vector r'(t) starts out at (12,-5,14) at t=0 and flies around for 1 second. Where is the bee located at time t=1 if ∫ from 0 to 1 r'(u) du = 0

A bee with a velocity vector r'(t) starts out at (12,-5,14) at t=0 and flies around for 1 second. Where is the bee located at time t=1 if

∫ from 0 to 1 r'(u) du = 0

Transcript text: A bee with a velocity vector $\mathbf{r}^{\prime}(t)$ starts out at $(12,-5,14)$ at $t=0$ and flies around for 1 seconds. Where is the bee located at time $t=1$ if \[ \int_{0}^{1} \mathbf{r}^{\prime}(u) d u=0 \]

Solution

Solution Steps

To find the location of the bee at time $ t=1 $, we need to use the given information about the integral of the velocity vector. The integral of the velocity vector over the interval from $ t=0 $ to $ t=1 $ gives the displacement of the bee. Since the integral is given to be zero, the displacement is zero, meaning the bee's position does not change over the interval.

Solution Approach

The initial position of the bee at $ t=0 $ is given as $ (12, -5, 14) $.
The integral of the velocity vector from $ t=0 $ to $ t=1 $ is zero, indicating no net displacement.
Therefore, the position of the bee at $ t=1 $ is the same as the initial position.

Step 1: Initial Position

The initial position of the bee at $ t=0 $ is given as $ (12, -5, 14) $.

Step 2: Displacement Calculation

The integral of the velocity vector from $ t=0 $ to $ t=1 $ is given by: \[ \int_{0}^{1} \mathbf{r}^{\prime}(u) \, du = 0 \] This indicates that the net displacement of the bee over the interval is zero.

Step 3: Final Position

Since the displacement is zero, the position of the bee at $ t=1 $ is the same as the initial position.