Q: After $$6$$ seconds, the force on the particle in the previous question changes to $$\mathbf{F} = \langle -4, -4, -2 \rangle (N). $$What is the acceleration vector immediately after the change?

$$\langle -4, -4, -2 \rangle\ \text{m/s}^2$$

Q: A $$1\ \text{kg}$$ particle is acted upon by $$\mathbf{F} = \langle 2, 4, 4 \rangle (N) $$for $$6$$ seconds, starting from rest at the origin. What is its position vector at $$t = 6 s$$?

$$\langle 36, 72, 72 \rangle\ \text{m}$$

Q: A particle’s position is $$\mathbf{r}(t) = \langle t^2$, 2t^2$, 2t^2$ \rangle (m) $$for $$0 \leq t \leq 6. $$What is its velocity vector at $$t = 6 s$$?

$$\langle 12, 24, 24 \rangle\ \text{m/s}$$

Q: A particle’s position is $$\mathbf{r}(t) = \langle -108 + 36t - 2t^2$, -144 + 48t - 2t^2$, -108 + 36t - t^2$ \rangle (m) $$for $$t \geq 6. $$What is its velocity vector?

$$\langle 36 - 4t, 48 - 4t, 36 - 2t \rangle\ \text{m/s}$$

Q: If a particle’s acceleration is $$\mathbf{a}(t) = \langle 2, 4, 4 \rangle (m/s$^2$$), what is the force acting on a $$1\ \text{kg}$$ particle?

$$\langle 2, 4, 4 \rangle\ \text{N}$$

Q: A $$2\ \text{kg}$$ particle is acted upon by $$\mathbf{F} = \langle 8, 0, 0 \rangle (N). $$What is its acceleration?

$$\langle 4, 0, 0 \rangle\ \text{m/s}^2$$

Question 1

A particle of mass $$2\ 	ext{kg} is $$acted upon by two forces: $$F^{(1)}$$ = \langle 5\ 	ext{N}, 3\ 	ext{N}, 0\ 	ext{N} angle$$ and F$$^{(2)}$$ = \langle a\ 	ext{N}, b\ 	ext{N}, 0\ 	ext{N} angle. If $$its position at time $$t is$$ $$r(t) = \langle 1 - t + 2t^2$, 2 + 4t, 5 angle (in $$meters), what are the values of $$a$$ and $$b$$?

Accepted Answer

$$a = 3,\ b = -3$$

Question 2

Which of the following best describes Newton’s Second Law of Motion in vector form?

Accepted Answer

$$m \dfrac{d^2$ \mathbf{r}}{dt^2$} = \mathbf{F}^{(1)} + \cdots + \mathbf{F}^{(n)}$$

Question 3

A $$1\ 	ext{kg}$$ particle is initially at $$\mathbf{r}(0) = \langle -1, -1, 0 angle (m) $$and $$\mathbf{v}(0) = \langle 4, 8, 0 angle (m/s). It is $$acted upon by $$\mathbf{F}^{(1)} = \langle 4, 0, 0 angle (N) $$and $$\mathbf{F}^{(2)} = \langle 0, 8, 0 angle (N). $$What is its position vector at time $$t$$?

Accepted Answer

$$\mathbf{r}(t) = \langle 2t^2$ + 4t - 1,\ 4t^2$ + 8t - 1,\ 0 \rangle$$

Question 4

A $$1\ 	ext{kg}$$ particle is at rest at the origin. A force $$\mathbf{F} = \langle 2, 4, 4 angle (N) $$acts for $$6$$ seconds. What is the particle’s speed at $$t = 6 s$$?

Accepted Answer

$$36\ 	ext{m/s}$$

Question 5

After $$6$$ seconds, the force on the particle in the previous question changes to $$\mathbf{F} = \langle -4, -4, -2 \rangle (N). $$What is the acceleration vector immediately after the change?

Accepted Answer

$$\langle -4, -4, -2 angle\ 	ext{m/s}^2$$

Question 6

A particle’s velocity is $$\mathbf{v}(t) = \langle 4t + 4, 8t + 8, 0 \rangle (m/s). $$What is the magnitude of its speed at $$t = 1 s$$?

Accepted Answer

$$\sqrt{80}\ 	ext{m/s}$$

Question 7

If a force of $$1\ 	ext{N}$$ acts on a $$1\ 	ext{kg}$$ mass, what is the resulting acceleration?

Accepted Answer

$$1\ 	ext{m/s}^2$$

Question 8

A particle moves in the $$xy-$$plane along a straight line described by $$y = 2x + 1. $$Which statement is true about the forces acting on the particle?

Accepted Answer

The forces only increase the magnitude of the velocity, not its direction.

Question 9

A $$1\ 	ext{kg}$$ particle is acted upon by $$\mathbf{F} = \langle 2, 4, 4 angle (N) $$for $$6$$ seconds, starting from rest at the origin. What is its position vector at $$t = 6 s$$?

Accepted Answer

$$\langle 36, 72, 72 angle\ 	ext{m}$$

Question 10

Given $$\mathbf{v}(t) = \langle 4t + 4, 8t + 8, 0 \rangle (m/s) $$and $$\mathbf{a}(t) = \langle 4, 8, 0 \rangle (m/s$^2$$), what is the tangential component of acceleration at $$t = 1 s$$?

Accepted Answer

$$\sqrt{80}\ 	ext{m/s}^2$$

Question 11

A particle’s acceleration vector is $$\mathbf{a}(t) = \langle 4, 8, 0 \rangle (m/s$^2). $$What is the normal component of acceleration?

Accepted Answer

$$0\ 	ext{m/s}^2$$

Question 12

A particle of mass $$m is $$acted upon by a constant force $$\mathbf{F}. $$Which equation describes its motion?

Accepted Answer

$$m \dfrac{d^2$ \mathbf{r}}{dt^2$} = \mathbf{F}$$

Question 13

A particle’s position is $$\mathbf{r}(t) = \langle t^2$, 2t^2$, 2t^2$ \rangle (m) $$for $$0 \leq t \leq 6. $$What is its velocity vector at $$t = 6 s$$?

Accepted Answer

$$\langle 12, 24, 24 angle\ 	ext{m/s}$$

Question 14

A particle’s position is $$\mathbf{r}(t) = \langle -108 + 36t - 2t^2$, -144 + 48t - 2t^2$, -108 + 36t - t^2$ \rangle (m) $$for $$t \geq 6. $$What is its velocity vector?

Accepted Answer

$$\langle 36 - 4t, 48 - 4t, 36 - 2t angle\ 	ext{m/s}$$

Question 15

If a particle’s acceleration is $$\mathbf{a}(t) = \langle 2, 4, 4 angle (m/s$^2$$), what is the force acting on a $$1\ 	ext{kg}$$ particle?

Accepted Answer

$$\langle 2, 4, 4 angle\ 	ext{N}$$

Question 16

A particle’s position is $$\mathbf{r}(t) = \langle 2t^2$ + 4t - 1, 4t^2$ + 8t - 1, 0 \rangle (m). $$What is the equation of the straight line along which it moves in the $$xy-$$plane?

Accepted Answer

$$y = 2x + 1$$

Question 17

A $$2\ 	ext{kg}$$ particle is acted upon by $$\mathbf{F} = \langle 8, 0, 0 angle (N). $$What is its acceleration?

Accepted Answer

$$\langle 4, 0, 0 angle\ 	ext{m/s}^2$$

Question 18

Which of the following is the correct SI unit for force?

Accepted Answer

$$	ext{kg}\cdot	ext{m}/	ext{s}^2$$

Newton’s Second Law of Motion: Vector Formulation and Applications

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