Q: A puck moves on an air hockey table with initial velocity components $$v_{ix} = +1.0\ \mathrm{m/s}$$ and $$v_{iy} = +2.0\ \mathrm{m/s}$$, and acceleration components $$a_x = +2.0\ \mathrm{m/s^2$}$$ and $$a_y = -2.0\ \mathrm{m/s^2$}. $$What is the magnitude of the puck's velocity after $$0.50\ \mathrm{s}$$?

$$\sqrt{(1.0 + 2.0 \times 0.5)^2$ + (2.0 - 2.0 \times 0.5)^2$} = \sqrt{(2.0)^2$ + (1.0)^2$} = \sqrt{5} \approx 2.24\ \mathrm{m/s}$$

Q: A projectile is launched from ground level with an initial speed $$v_0 at an $$angle $$\theta$$ above the horizontal. Neglecting air resistance, what is the formula for the maximum height $$h_{max}$$ reached by the projectile?

$$h_{max} = \frac{v_0$^2 \sin^2$ \theta}{2g}$$

Q: Which of the following is the correct expression for the instantaneous velocity vector $$\vec{v} in $$two dimensions?

$$\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t}$$

Q: A golfer hits a ball with a speed of $$14.0\ \mathrm{m/s} at an $$angle of $$40.0^\circ$$ above the horizontal from a point $$3.0\ \mathrm{m}$$ above the green. Ignoring air resistance, what is the time $$t it $$takes for the ball to reach the green?

$$t = \frac{v_0$ \sin \theta + \sqrt{(v_0$ \sin \$$theta)^2$$ + 2g h}}{g}$$

Question 1

Which of the following statements best describes the independence of horizontal and vertical motions in two-dimensional kinematics?

Accepted Answer

Horizontal and vertical motions are independent of each other and always perpendicular (at 90 degrees).

Question 2

Given the position vector $$\vec{r} of an $$object in two dimensions, which equation correctly expresses the displacement vector $$\Delta \vec{r}$$ between two points?

Accepted Answer

$$\Delta \vec{r} = \vec{r}_f - \vec{r}_i$$

Question 3

A puck moves on an air hockey table with initial velocity components $$v_{ix} = +1.0\ \mathrm{m/s}$$ and $$v_{iy} = +2.0\ \mathrm{m/s}$$, and acceleration components $$a_x = +2.0\ \mathrm{m/s^2$}$$ and $$a_y = -2.0\ \mathrm{m/s^2$}. $$What is the magnitude of the puck's velocity after $$0.50\ \mathrm{s}$$?

Accepted Answer

$$\sqrt{(1.0 + 2.0 	imes 0.5)^2$ + (2.0 - 2.0 	imes 0.5)^2$} = \sqrt{(2.0)^2$ + (1.0)^2$} = \sqrt{5} \approx 2.24\ \mathrm{m/s}$$

Question 4

Which equation correctly defines the average velocity vector $$\vec{v}_{avg} in $$two dimensions?

Accepted Answer

$$\vec{v}_{avg} = \frac{\vec{r}_f - \vec{r}_i}{t_f - t_i}$$

Question 5

A projectile is launched from ground level with an initial speed $$v_0 at an $$angle $$	heta$$ above the horizontal. Neglecting air resistance, what is the formula for the maximum height $$h_{max}$$ reached by the projectile?

Accepted Answer

$$h_{max} = \frac{v_0$^2 \sin^2$ 	heta}{2g}$$

Question 6

Which of the following best describes when an object is accelerating in two dimensions?

Accepted Answer

When either the magnitude or the direction of the velocity vector changes.

Question 7

A marble is thrown horizontally from the top of a building with a speed of $$15\ \mathrm{m/s}. $$When it strikes the ground, its velocity makes an angle of $$65^\circ$$ with the horizontal. From what height was the marble thrown?

Accepted Answer

Question 8

Which of the following is the correct expression for the instantaneous velocity vector $$\vec{v} in $$two dimensions?

Accepted Answer

$$\vec{v} = \lim_{\Delta t 	o 0} \frac{\Delta \vec{r}}{\Delta t}$$

Question 9

A golfer hits a ball with a speed of $$14.0\ \mathrm{m/s} at an $$angle of $$40.0^\circ$$ above the horizontal from a point $$3.0\ \mathrm{m}$$ above the green. Ignoring air resistance, what is the time $$t it $$takes for the ball to reach the green?

Accepted Answer

$$t = \frac{v_0$ \sin 	heta + \sqrt{(v_0$ \sin \$$theta)^2$$ + 2g h}}{g}$$

Question 10

Which of the following is NOT an assumption made in the analysis of projectile motion?

Accepted Answer

The rotation of the Earth affects the motion.

Question 11

An airplane is climbing upward at $$97.5\ \mathrm{m/s} at an $$angle of $$50.0^\circ$$ above the horizontal. When the plane is at an altitude of $$732\ \mathrm{m}$$, a package is released. What is the initial vertical component of the package's velocity?

Accepted Answer

$$v_{iy} = 97.5 \sin 50.0^\circ$$

Question 12

Which of the following is the correct kinematic equation for the $$x-$$component of velocity under constant acceleration?

Accepted Answer

$$v_{fx} = v_{ix} + a_x \Delta t$$

Question 13

A skateboarder shoots off a ramp with a velocity of $$6.6\ \mathrm{m/s} at an $$angle of $$58^\circ$$ above the horizontal. The end of the ramp is $$1.2\ \mathrm{m}$$ above the ground. What is the vertical component of the initial velocity?

Accepted Answer

$$v_{iy} = 6.6 \sin 58^\circ$$

Question 14

Which of the following is the correct equation for the $$y-$$component of displacement under constant acceleration?

Accepted Answer

$$y_f = y_i + v_{iy} \Delta t + \frac{1}{2} a_y (\Delta t)^2$

Question 15

A projectile is launched with initial velocity $$v_0 at $$angle $$	heta$$ above the horizontal. What is the total time of flight if it lands at the same vertical level from which it was launched?

Accepted Answer

$$t = \frac{2 v_0$ \sin 	heta}{g}$$

Question 16

Which of the following is the correct expression for the average acceleration vector $$\vec{a}_{avg} in $$two dimensions?

Accepted Answer

$$\vec{a}_{avg} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i}$$

Question 17

A projectile is launched at $$v_0$$ and angle $$	heta$$ above the horizontal. What is the formula for the horizontal range $$R ($$assuming it lands at the same height)?

Accepted Answer

$$R = \frac{v_0$^2 \sin 2	heta}{g}$$

Question 18

When solving a two-dimensional kinematics problem, which of the following is NOT a recommended step?

Accepted Answer

Ignore the time interval and solve for displacement directly.

Kinematics in Two Dimensions: Position, Displacement, Velocity, Acceleration, and Projectile Motion

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