Question 1

A roller coaster car moves in a vertical loop as shown in the first image. At the top of the loop, which of the following statements about the acceleration of the car is correct?

Accepted Answer

The acceleration points toward the center of the loop.

Question 2

Given a position vector in three dimensions $$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$, which of the following correctly expresses the displacement vector $$\Delta \vec{r}$$ between two points $$\vec{r}_1$$ and $$\vec{r}_2$$?

Accepted Answer

$$\Delta \vec{r} = (x_2$ - x_1$)\hat{i} + (y_2$ - y_1$)\hat{j} + (z_2$ - z_1$)\hat{k}$$

Question 3

A particle moves from $$\vec{r}_1 = (2.0\ \mathrm{m}, 1.0\ \mathrm{m}, 0.0\ \mathrm{m}) to$$ $$\vec{r}_2 = (3.0\ \mathrm{m}, 5.0\ \mathrm{m}, -4.0\ \mathrm{m}) in$$ $$4.0\ \mathrm{s}. $$What is the average velocity vector $$\vec{v}_{ave}$$?

Accepted Answer

$$\vec{v}_{ave} = (0.25\hat{i} + 1.0\hat{j} - 1.0\hat{k})\ \mathrm{m/s}$$

Question 4

Which of the following best describes the direction of the instantaneous velocity vector $$\vec{v}$$ for a particle moving along a curved path?

Accepted Answer

It is always tangent to the path.

Question 5

A car is rounding a curve and slowing down, as shown in the images. Which of the following statements is true about its acceleration?

Accepted Answer

The acceleration has both a component tangent to the path and a component perpendicular to the path.

Question 6

For a particle in uniform circular motion of radius $$R$$ and speed $$v$$, what is the magnitude of the acceleration?

Accepted Answer

$$a = \dfrac{v^2$}{R}$$

Question 7

A bird flies in the $$xy-$$plane with velocity $$\vec{v}(t) = (2.4 - 1.6t^2$)\hat{i} + (4.0t)\hat{j} (in m/s). At $$what time $$t \u003e 0 is $$the bird moving purely in the $$y-$$direction?

Accepted Answer

$$t = 1.2\ \mathrm{s}$$

Question 8

The period $$T of a $$particle moving at constant speed $$v$$ around a circle of radius $$R is $$given by:

Accepted Answer

$$T = 2\pi R / v$$

Question 9

A car tire can withstand a maximum centripetal acceleration of $$9.8\ \mathrm{m/s^2$}$$ ("1 g"). What is the minimum radius of a turn the car can make at $$27\ \mathrm{m/s}$$?

Accepted Answer

$$R = 74.5\ \mathrm{m}$$

Question 10

Which of the following is true for the average acceleration vector $$\vec{a}_{ave}$$ between two points $$P_1$$ and $$P_2 on a $$curved path?

Accepted Answer

It points in the direction of the change in velocity $$\Delta \vec{v}.$$

Question 11

A projectile is launched horizontally from a cliff. Which statement is correct about the time it takes to hit the ground compared to a projectile simply dropped from the same height?

Accepted Answer

Both projectiles hit the ground at the same time.

Question 12

If a particle's velocity vector changes direction but not magnitude, what can be said about its acceleration?

Accepted Answer

The acceleration is perpendicular to the velocity.

Question 13

A particle moves in three dimensions with position $$\vec{r}(t) = (3t^2$)\hat{i} + (2t)\hat{j} + (4)\hat{k}. $$What is the instantaneous velocity at $$t = 1\ \mathrm{s}$$?

Accepted Answer

$$\vec{v} = (6\hat{i} + 2\hat{j})\ \mathrm{m/s}$$

Question 14

Which of the following best describes the parallel and perpendicular components of acceleration for a particle moving along a curved path?

Accepted Answer

The parallel component changes the speed, and the perpendicular component changes the direction of motion.

Question 15

A projectile is launched at an angle in the $$xy-$$plane. Which of the following is true about its acceleration (neglecting air resistance)?

Accepted Answer

The acceleration is always downward and constant.

Question 16

A particle moves in a circle of radius $$R$$ with increasing speed. Which statement is correct about its acceleration?

Accepted Answer

It has both radial and tangential components.

Question 17

If the velocity of a particle is given by $$\vec{v}(t) = (3t)\hat{i} + (4)\hat{j}$$, what is the magnitude of the acceleration?

Accepted Answer

$$a = 3\ \mathrm{m/s^2$}$$

Motion in Two and Three Dimensions: Physics with Calculus Study Notes

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