Q: A soccer ball is kicked at $$20\ \mathrm{m/s} at an $$angle of $$35^\circ. $$What is the initial vertical velocity component?

$$v_{0y} = v_0$ \sin \theta = 20 \sin 35^\circ = 11.47\ \mathrm{m/s}$$

Q: A car is driving in a circle of radius $$6\ \mathrm{m}. If $$the maximum acceleration before sliding is $$8\ \mathrm{m/s^2$}$$, what is the maximum speed the car can have?

$$v_{max} = \sqrt{a_{max} R} = \sqrt{8 \times 6} = 6.93\ \mathrm{m/s}$$

Q: A quarterback wants to throw a football $$35\ \mathrm{m}$$ downfield with a maximum speed of $$20\ \mathrm{m/s}. $$What launch angle should he use?

$$\theta = \dfrac{1}{2} \sin$$^{-1}$$ \left( \dfrac{gR}{v_0$^2} \right) = 29.5^\circ$$

Q: A projectile is launched with an initial speed $$v_0 at an $$angle $$\theta$$ above the horizontal. What is the formula for the range $$R on $$level ground?

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

Q: A projectile is launched upward with $$v_{0y} = 15\ \mathrm{m/s}. $$What is the maximum height it reaches? (Use $$g = 9.8\ \mathrm{m/s^2$}.)$$

$$y_{max} = \dfrac{v_{0y}^2}{2g} = \dfrac{(15)^2$}{2 \times 9.8} = 11.5\ \mathrm{m}$$

Question 1

A football is thrown at a $$45^\circ$$ angle as shown in the image. If the initial speed is $$22.3\ \mathrm{m/s}$$, what is the maximum range the football can achieve on level ground? (Assume $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$R = \dfrac{v_0$^2}{g} = \dfrac{(22.3)^2$}{9.8} = 50.8\ \mathrm{m}$$

Question 2

At the highest point of a projectile's trajectory, which of the following statements is true?

Accepted Answer

The vertical velocity is zero.

Question 3

A soccer ball is kicked at $$20\ \mathrm{m/s} at an $$angle of $$35^\circ. $$What is the initial vertical velocity component?

Accepted Answer

$$v_{0y} = v_0$ \sin 	heta = 20 \sin 35^\circ = 11.47\ \mathrm{m/s}$$

Question 4

Which of the following best describes the acceleration of a projectile in flight (neglecting air resistance)?

Accepted Answer

Constant and downward.

Question 5

A car is driving in a circle of radius $$6\ \mathrm{m}. If $$the maximum acceleration before sliding is $$8\ \mathrm{m/s^2$}$$, what is the maximum speed the car can have?

Accepted Answer

$$v_{max} = \sqrt{a_{max} R} = \sqrt{8 	imes 6} = 6.93\ \mathrm{m/s}$$

Question 6

If a projectile is launched with a vertical velocity $$v_{0y}$$, what is the time to reach the highest point?

Accepted Answer

$$t_{top} = \dfrac{v_{0y}}{g}$$

Question 7

A quarterback wants to throw a football $$35\ \mathrm{m}$$ downfield with a maximum speed of $$20\ \mathrm{m/s}. $$What launch angle should he use?

Accepted Answer

$$	heta = \dfrac{1}{2} \sin$$^{-1}$$ \left( \dfrac{gR}{v_0$^2} ight) = 29.5^\circ$$

Question 8

Which of the following is true for the horizontal velocity of a projectile (neglecting air resistance)?

Accepted Answer

It remains constant throughout the flight.

Question 9

A projectile is launched and returns to the same altitude. Which of the following is true about its speed when it lands?

Accepted Answer

It lands at the same speed it was launched (but downward).

Question 10

For an object moving in a circle of radius $$R$$ with constant speed $$v$$, what is the magnitude of the centripetal acceleration?

Accepted Answer

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

Question 11

A projectile is launched with an initial speed $$v_0 at an $$angle $$	heta$$ above the horizontal. What is the formula for the range $$R on $$level ground?

Accepted Answer

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

Question 12

If the period of revolution for an object in circular motion is $$T$$, what is its angular velocity $$\omega$$?

Accepted Answer

$$\omega = \dfrac{2\pi}{T}$$

Question 13

A projectile is launched upward with $$v_{0y} = 15\ \mathrm{m/s}. $$What is the maximum height it reaches? (Use $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$y_{max} = \dfrac{v_{0y}^2}{2g} = \dfrac{(15)^2$}{2 	imes 9.8} = 11.5\ \mathrm{m}$$

Question 14

If a projectile spends $$t$$ seconds going up, how long does it spend coming down (to the same altitude)?

Accepted Answer

It spends the same amount of time coming down as going up.

Projectile Motion and Circular Motion: Range, Trajectories, and Applications

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