Q: Given the position vector $$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$, which of the following correctly expresses the average velocity vector $$\vec{v}_{\text{ave}}$$ between two points $$P_1$$ and $$P_2 at $$times $$t_1$$ and $$t_2$$?

$$\vec{v}_{\text{ave}} = \dfrac{\vec{r}_2 - \vec{r}_1}{t_2$ - t_1$}$$

Q: If a particle's position changes from $$(1,2,3) m to$$ $$(4,6,3) m in 2 $$seconds, what is its average velocity vector?

$$\vec{v}_{\text{ave}} = 1.5\hat{i} + 2\hat{j} m/s$$

Question 1

Given the position vector $$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$, which of the following correctly expresses the average velocity vector $$\vec{v}_{	ext{ave}}$$ between two points $$P_1$$ and $$P_2 at $$times $$t_1$$ and $$t_2$$?

Accepted Answer

$$\vec{v}_{	ext{ave}} = \dfrac{\vec{r}_2 - \vec{r}_1}{t_2$ - t_1$}$$

Question 2

Which statement best describes the direction of the instantaneous velocity vector at any point along a particle's path?

Accepted Answer

It is tangent to the path at that point.

Question 3

If a particle's position is given by $$\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$$, what is the correct expression for its instantaneous velocity?

Accepted Answer

$$\vec{v} = \dfrac{d\vec{r}}{dt} = \dfrac{dx}{dt}\hat{i} + \dfrac{dy}{dt}\hat{j} + \dfrac{dz}{dt}\hat{k}$$

Question 4

A particle moves such that $$x(t) = 2t$$, $$y(t) = 3t^2$, and z(t) = 4$. What is the instantaneous velocity vector at t = 1$ s?

Accepted Answer

$$\vec{v} = 2\hat{i} + 6\hat{j} + 0\hat{k}$$

Question 5

Which of the following is NOT a correct property of the average velocity vector?

Accepted Answer

It is always tangent to the path.

Question 6

If a particle's position changes from $$(1,2,3) m to$$ $$(4,6,3) m in 2 $$seconds, what is its average velocity vector?

Accepted Answer

$$\vec{v}_{	ext{ave}} = 1.5\hat{i} + 2\hat{j} m/s$$

Question 7

Which mathematical operation is used to find the instantaneous velocity from the position vector?

Accepted Answer

Differentiation with respect to time

Question 8

A particle moves along a path described by $$\vec{r}(t) = t^2$\hat{i} + t^3$\hat{j}. $$What is the magnitude of its instantaneous velocity at $$t = 2 s$$?

Accepted Answer

$$\sqrt{16 + 144} m/s$$

Question 9

If $$\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$$, which of the following correctly expresses the acceleration vector?

Accepted Answer

$$\vec{a} = \dfrac{d^2$x}{dt^2$}\hat{i} + \dfrac{d^2$y}{dt^2$}\hat{j} + \dfrac{d^2$z}{dt^2$}\hat{k}$$

Question 10

A car moves along a straight path such that its position is given by $$x(t) = 5t^2$. What is its instantaneous velocity at t = 3$ s?

Accepted Answer

$$30 m/s$$

Question 11

Which of the following statements is TRUE regarding the vector nature of velocity?

Accepted Answer

Velocity has both magnitude and direction.

Question 12

If a particle's instantaneous velocity is zero at a certain point, which of the following must be true?

Accepted Answer

The particle is momentarily at rest at that point.

Question 13

A particle moves such that $$x(t) = t$$, $$y(t) = t^2$, z(t$$) = $$t^3. $$What is the instantaneous velocity vector at $$t = 2 s$$?

Accepted Answer

$$1\hat{i} + 4\hat{j} + 12\hat{k}$$

Question 14

Which of the following best describes the relationship between the position, velocity, and acceleration vectors?

Accepted Answer

Velocity is the time derivative of position, and acceleration is the time derivative of velocity.

Question 15

A particle's position is given by $$\vec{r}(t) = 3t\hat{i} + 2t^2$\hat{j}. $$What is the acceleration vector at $$t = 1 s$$?

Accepted Answer

$$0\hat{i} + 4\hat{j}$$

Question 16

If the instantaneous velocity vector is always perpendicular to the position vector, what type of motion does the particle exhibit?

Accepted Answer

Uniform circular motion

Question 17

A particle moves along a path such that $$\vec{r}(t) = t\hat{i} + t^2$\hat{j}. $$What is the equation for the tangent to the path at $$t = 1 s$$?

Accepted Answer

$$\vec{r}_{	ext{tan}} = (1 + \lambda)\hat{i} + (1 + 2\lambda)\hat{j}$$

Kinematics in Three Dimensions: Position, Velocity, and Instantaneous Velocity

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