Q: A ball is thrown horizontally from the top of a 45 m high cliff with an initial speed of 10 m/s. How far from the base of the cliff will the ball land? (Assume $$g = 9.81\ \mathrm{m/s^2$}$$ and ignore air resistance.)

$$x = v_{0x} \sqrt{\frac{2h}{g}} = 10 \sqrt{\frac{2 \times 45}{9.81}} \approx 30.3\ \mathrm{m}$$

Q: A diver jumps off a platform with an initial velocity of $$5\ \mathrm{m/s} at an $$angle of $$0^\circ ($$horizontal) from a height of $$10\ \mathrm{m}. $$How long does it take for the diver to hit the water?

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 10}{9.81}} \approx 1.43\ \mathrm{s}$$

Q: For a projectile launched from the ground with initial speed $$v_0 at $$angle $$\theta$$, what is the formula for the range $$R if $$the landing elevation equals the launch elevation?

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

Q: A firework is launched at $$45^\circ$$ with a speed of $$30\ \mathrm{m/s}. $$What is the total time the firework spends in the air? (Assume it lands at the same elevation it was launched.)

$$t = \frac{2 v_0$ \sin \theta}{g} = \frac{2 \times 30 \times \sin 45^\circ}{9.81} \approx 4.33\ \mathrm{s}$$

Q: A water fountain shoots water at an angle of $$35^\circ$$ with a speed of $$8\ \mathrm{m/s}. $$What is the horizontal component of the initial velocity?

$$v_{0x} = v_0$ \cos \theta = 8 \cos 35^\circ \approx 6.56\ \mathrm{m/s}$$

Q: A projectile is launched from the ground at $$20\ \mathrm{m/s} at$$ $$60^\circ. $$What is its vertical velocity when it returns to the ground?

$$v_{y} = -v_0$ \sin \theta = -20 \sin 60^\circ \approx -17.3\ \mathrm{m/s}$$

Question 1

A ball is thrown horizontally from the top of a 45 m high cliff with an initial speed of 10 m/s. How far from the base of the cliff will the ball land? (Assume $$g = 9.81\ \mathrm{m/s^2$}$$ and ignore air resistance.)

Accepted Answer

$$x = v_{0x} \sqrt{\frac{2h}{g}} = 10 \sqrt{\frac{2 	imes 45}{9.81}} \approx 30.3\ \mathrm{m}$$

Question 2

Which of the following best describes the acceleration of a projectile in two-dimensional motion (ignoring air resistance)?

Accepted Answer

$$a_x = 0,\ a_y = -g$$

Question 3

A soccer ball is kicked from the ground with an initial speed of $$20\ \mathrm{m/s} at an $$angle of $$30^\circ$$ above the horizontal. What is the maximum height reached by the ball? (Use $$g = 9.81\ \mathrm{m/s^2$})$$

Accepted Answer

$$h_{max} = \frac{(v_0$ \sin \$$theta)^2$$}{2g} = \frac{(20 \sin 30^\$$circ)^2$$}{2 	imes 9.81} \approx 5.1\ \mathrm{m}$$

Question 4

Which of the following statements about projectile motion is correct?

Accepted Answer

The horizontal and vertical motions are independent except for sharing the same time of flight.

Question 5

A diver jumps off a platform with an initial velocity of $$5\ \mathrm{m/s} at an $$angle of $$0^\circ ($$horizontal) from a height of $$10\ \mathrm{m}. $$How long does it take for the diver to hit the water?

Accepted Answer

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 	imes 10}{9.81}} \approx 1.43\ \mathrm{s}$$

Question 6

For a projectile launched from the ground with initial speed $$v_0 at $$angle $$	heta$$, what is the formula for the range $$R if $$the landing elevation equals the launch elevation?

Accepted Answer

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

Question 7

A firework is launched at $$45^\circ$$ with a speed of $$30\ \mathrm{m/s}. $$What is the total time the firework spends in the air? (Assume it lands at the same elevation it was launched.)

Accepted Answer

$$t = \frac{2 v_0$ \sin 	heta}{g} = \frac{2 	imes 30 	imes \sin 45^\circ}{9.81} \approx 4.33\ \mathrm{s}$$

Question 8

Which launch angle will give a projectile the maximum range (assuming the same initial speed and launch/landing elevation)?

Accepted Answer

$$45^\circ$$

Question 9

A water fountain shoots water at an angle of $$35^\circ$$ with a speed of $$8\ \mathrm{m/s}. $$What is the horizontal component of the initial velocity?

Accepted Answer

$$v_{0x} = v_0$ \cos 	heta = 8 \cos 35^\circ \approx 6.56\ \mathrm{m/s}$$

Question 10

A projectile is launched from the ground at $$20\ \mathrm{m/s} at$$ $$60^\circ. $$What is its vertical velocity when it returns to the ground?

Accepted Answer

$$v_{y} = -v_0$ \sin 	heta = -20 \sin 60^\circ \approx -17.3\ \mathrm{m/s}$$

Question 11

Which of the following real-world examples best illustrates projectile motion?

Accepted Answer

A fountain jet arcing through the air before landing in a pool.

Question 12

If a projectile is launched horizontally from a height $$h$$ with speed $$v_0$$, what is the equation for its trajectory $$y as a $$function of $$x$$?

Accepted Answer

$$y = h - \frac{g}{2 v_0$^2} x^2$

Question 13

A ball is thrown at $$25^\circ$$ above the horizontal with a speed of $$15\ \mathrm{m/s}. $$What is the initial vertical velocity?

Accepted Answer

$$v_{0y} = v_0$ \sin 	heta = 15 \sin 25^\circ \approx 6.34\ \mathrm{m/s}$$

Question 14

Which of the following is NOT a characteristic of projectile motion (ignoring air resistance)?

Accepted Answer

The horizontal acceleration is nonzero.

Question 15

A student throws a rock horizontally from a bridge $$20\ \mathrm{m}$$ above a river with a speed of $$7\ \mathrm{m/s}. $$How long does it take to hit the water?

Accepted Answer

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 	imes 20}{9.81}} \approx 2.02\ \mathrm{s}$$

Question 16

A projectile is launched at $$v_0$$ and angle $$	heta. $$Which equation gives the vertical position $$y as a $$function of time $$t$$?

Accepted Answer

$$y = (v_0$ \sin 	heta)t - \frac{1}{2}gt^2$

Two-Dimensional Kinematics and Projectile Motion

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