Question 1

Given the diagram above, which of the following statements best describes the relationship between $$\vec{r}(t)$$ and $$\vec{v}(t) at $$any time $$t$$?

Accepted Answer

$$\vec{v}(t) is $$always tangent to the trajectory $$\vec{r}(t) at $$each point.

Question 2

Suppose a particle moves along the trajectory $$\vec{r}(t) as $$shown in the diagram. If $$\vec{v}(t_2$) is $$longer than $$\vec{v}(t_1$)$$, what does this indicate about the particle's motion between $$t_1$$ and $$t_2$$?

Accepted Answer

The particle is speeding up between $$t_1$$ and $$t_2.$$

Question 3

If $$\vec{v}(t_3$) is $$perpendicular to the $$z-$$axis at $$t_3$$, what does this tell you about the particle's motion at that instant?

Accepted Answer

The particle is not moving in the $$z-$$direction at $$t_3.$$

Question 4

Which of the following best describes the physical meaning of $$\vec{r}'(t) in $$the context of the diagram?

Accepted Answer

$$\vec{r}'(t)$$ represents the velocity vector of the particle at time $$t.$$

Question 5

If the particle's speed is increasing at $$t_2$$, which of the following must be true about the acceleration $$\vec{a}(t_2$)$$?

Accepted Answer

$$\vec{a}(t_2$)$$ has a component in the direction of $$\vec{v}(t_2$).$$

Question 6

Suppose $$\vec{r}(t)$$ describes the position of a drone in 3D space. If $$\vec{v}(t_1$)$$ points directly upward along the $$z-$$axis, what is the drone doing at $$t_1$$?

Accepted Answer

The drone is moving vertically upward at $$t_1.$$

Question 7

If $$\vec{r}(t_2$)$$ and $$\vec{v}(t_2$)$$ are not parallel, what does this indicate about the particle's motion at $$t_2$$?

Accepted Answer

The particle is not moving directly away from or toward the origin at $$t_2.$$

Question 8

Which of the following statements is true about the trajectory $$\vec{r}(t)$$ shown in the diagram?

Accepted Answer

The trajectory $$\vec{r}(t)$$ can represent the path of a particle moving in three-dimensional space.

Question 9

If the velocity vectors $$\vec{v}(t_1$)$$, $$\vec{v}(t_2$)$$, and $$\vec{v}(t_3$)$$ have different directions, what does this imply about the motion of the particle?

Accepted Answer

The particle is changing direction as it moves along its path.

Question 10

If $$\vec{v}(t_2$) is $$tangent to the curve at $$\vec{r}(t_2$)$$, which calculus concept does this illustrate?

Accepted Answer

The derivative of the position vector gives the tangent (velocity) vector.

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