Q: A vector $$\vec{A}$$ makes an angle $$\theta$$ with the x-axis. What are its components?

$$A_x = A \cos \theta$$, $$A_y = A \sin \theta$$

Q: A projectile is launched from ground level with initial velocity $$v_0 at $$angle $$\$$theta_0$$. What is the equation for its trajectory (path) y(x$$)$$?

$$y = (\tan \$$theta_0$$)x - \dfrac{g x^2$}{2 (v_0$ \cos \$$theta_0$$)^2$$}$

Question 1

Which of the following is a vector quantity?

Accepted Answer

Acceleration

Question 2

A displacement vector in 3D is given by $$\vec{r} = 3\hat{i} + 4\hat{j} + 12\hat{k}. $$What is the magnitude of this vector?

Accepted Answer

$$\sqrt{3^2$ + 4^2$ + 12^2$} = 13$$

Question 3

A rural mail carrier drives 22.0 km north, then 47.0 km in a direction 60° south of east. What is the x-component of her total displacement?

Accepted Answer

$$47.0 \cos 60^\circ = 23.5 km$$

Question 4

Which graphical method is INCORRECT for vector addition?

Accepted Answer

Adding magnitudes directly without considering direction

Question 5

Given two vectors of length 3.0 units each, what is the range of possible lengths for their sum?

Accepted Answer

From 0 to 6.0 units, but the minimum is actually $$|3.0 - 3.0| = 0$$ and the maximum is $$3.0 + 3.0 = 6.0$$ units

Question 6

A projectile is launched with initial speed $$v_0 at an $$angle $$\$$theta_0$$ above the horizontal. What is the equation for its horizontal range R$ when it lands at the same height?

Accepted Answer

$$R = \dfrac{v_0$^2 \sin 2\$$theta_0$$}{g}$$

Question 7

Which statement about projectile motion is correct?

Accepted Answer

The vertical motion has constant acceleration, while the horizontal motion has constant velocity.

Question 8

A car moves at constant speed around a circle. What is true about its acceleration?

Accepted Answer

It has only normal (centripetal) acceleration.

Question 9

If $$\vec{v}_1 = 4\hat{i} + 3\hat{j}$$ and $$\vec{v}_2 = -2\hat{i} + 5\hat{j}$$, what is $$\vec{v}_1 + \vec{v}_2$$?

Accepted Answer

$$2\hat{i} + 8\hat{j}$$

Question 10

Which formula gives the instantaneous velocity vector in 3D?

Accepted Answer

$$\vec{v} = \dfrac{d\vec{r}}{dt}$$

Question 11

A vector $$\vec{A}$$ makes an angle $$	heta$$ with the x-axis. What are its components?

Accepted Answer

$$A_x = A \cos 	heta$$, $$A_y = A \sin 	heta$$

Question 12

A projectile is launched from ground level with initial velocity $$v_0 at $$angle $$\$$theta_0$$. What is the equation for its trajectory (path) y(x$$)$$?

Accepted Answer

$$y = (	an \$$theta_0$$)x - \dfrac{g x^2$}{2 (v_0$ \cos \$$theta_0$$)^2$$}$

Question 13

Which of the following is NOT a property of unit vectors?

Accepted Answer

They have variable magnitude depending on the vector.

Question 14

A ball is thrown horizontally from a height of 8.0 m and lands 8.5 m away. What is the time taken to hit the ground? (Ignore air resistance, $$g = 9.8 m/s$^2)$$

Accepted Answer

$$t = \sqrt{\dfrac{2h}{g}} = \sqrt{\dfrac{2 	imes 8.0}{9.8}} \approx 1.28 s$$

Question 15

Which equation correctly expresses the average acceleration vector in 3D?

Accepted Answer

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

Question 16

A vector $$\vec{A} is $$subtracted from $$\vec{B}$$ ($$\vec{B} - \vec{A}). $$Which graphical method is used?

Accepted Answer

Reverse $$\vec{A}$$ and add to $$\vec{B}$$ using tail-to-head method

Question 17

A particle moves along a curved path. At any instant, its velocity vector points:

Accepted Answer

Along the tangent to the path

Question 18

Which of the following is true about tangential and normal acceleration?

Accepted Answer

Tangential acceleration changes the speed, normal acceleration changes the direction.

Vectors and Motion in Two and Three Dimensions: Study Notes for Chapter 3 (Part 1)

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