Q: A ball is thrown horizontally from the top of a 45 m high cliff with an initial speed of 5.0 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.)

$$x = v_{0x} t = 5.0 \times \sqrt{\frac{2 \times 45}{9.81}} = 5.0 \times 3.03 = 15.15\ \mathrm{m}$$

Q: A projectile is launched from the ground with an initial speed $$v_0 at an $$angle $$\theta$$ above the horizontal. Which of the following expressions gives the maximum range $$R ($$assuming the projectile lands at the same height it was launched)?

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

Q: A helicopter flies horizontally at $$11\ \mathrm{m/s}$$ and then begins to accelerate vertically upward at $$0.86\ \mathrm{m/s^2$}. $$What is the vertical displacement after $$5.3\ \mathrm{s}$$?

$$y = \frac{1}{2} a t^2$ = \frac{1}{2} \times 0.86 \times (5.3)^2$ = 12.08\ \mathrm{m}$$

Q: A mountain climber jumps horizontally across a crevasse that is $$2.74\ \mathrm{m}$$ lower and $$4.10\ \mathrm{m}$$ wide. What is the minimum initial speed needed to safely cross the crevasse? ($$g = 9.81\ \mathrm{m/s^2$})$$

$$v_0$$ = \frac{w}{\sqrt{2h/g}} = \frac{4.10}{\sqrt{2 \times 2.74 / 9.81}} = 5.0\ \mathrm{m/s}$$

Q: A projectile is launched at $$v_0 = 20.0$$\ \mathrm{m/s}$$ and $$\theta = 35.0^\circ$$. What is the y$-component of its velocity at t = 1.00$$\ \mathrm{s}$$? (g = 9.81$$\ \mathrm{$$m/s^2$$}$$)

$$v_y = v_0$ \sin \theta - gt = 20.0 \sin 35.0^\circ - 9.81 \times 1.00 = 11.47 - 9.81 = 1.66\ \mathrm{m/s}$$

Q: A person skateboarding at $$1.30\ \mathrm{m/s}$$ releases a ball from a height of $$1.25\ \mathrm{m}. $$How long does it take for the ball to hit the ground? ($$g = 9.81\ \mathrm{m/s^2$})$$

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 1.25}{9.81}} = 0.505\ \mathrm{s}$$

Question 1

A ball is thrown horizontally from the top of a 45 m high cliff with an initial speed of 5.0 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

$$x = v_{0x} t = 5.0 	imes \sqrt{\frac{2 	imes 45}{9.81}} = 5.0 	imes 3.03 = 15.15\ \mathrm{m}$$

Question 2

A projectile is launched from the ground with an initial speed $$v_0 at an $$angle $$	heta$$ above the horizontal. Which of the following expressions gives the maximum range $$R ($$assuming the projectile lands at the same height it was launched)?

Accepted Answer

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

Question 3

A helicopter flies horizontally at $$11\ \mathrm{m/s}$$ and then begins to accelerate vertically upward at $$0.86\ \mathrm{m/s^2$}. $$What is the vertical displacement after $$5.3\ \mathrm{s}$$?

Accepted Answer

$$y = \frac{1}{2} a t^2$ = \frac{1}{2} 	imes 0.86 	imes (5.3)^2$ = 12.08\ \mathrm{m}$$

Question 4

A mountain climber jumps horizontally across a crevasse that is $$2.74\ \mathrm{m}$$ lower and $$4.10\ \mathrm{m}$$ wide. What is the minimum initial speed needed to safely cross the crevasse? ($$g = 9.81\ \mathrm{m/s^2$})$$

Accepted Answer

$$v_0$$ = \frac{w}{\sqrt{2h/g}} = \frac{4.10}{\sqrt{2 	imes 2.74 / 9.81}} = 5.0\ \mathrm{m/s}$$

Question 5

A projectile is launched at $$v_0 = 20.0$$\ \mathrm{m/s}$$ and $$	heta = 35.0^\circ$$. What is the y$-component of its velocity at t = 1.00$$\ \mathrm{s}$$? (g = 9.81$$\ \mathrm{$$m/s^2$$}$$)

Accepted Answer

$$v_y = v_0$ \sin 	heta - gt = 20.0 \sin 35.0^\circ - 9.81 	imes 1.00 = 11.47 - 9.81 = 1.66\ \mathrm{m/s}$$

Question 6

A person skateboarding at $$1.30\ \mathrm{m/s}$$ releases a ball from a height of $$1.25\ \mathrm{m}. $$How long does it take for the ball to hit the ground? ($$g = 9.81\ \mathrm{m/s^2$})$$

Accepted Answer

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 	imes 1.25}{9.81}} = 0.505\ \mathrm{s}$$

Question 7

A projectile is launched from the ground at an angle of $$30^\circ. At $$what point in its trajectory does it have the least speed?

Accepted Answer

At the highest point in its flight

Question 8

A ball is dropped from rest and another is fired horizontally from the same height at the same time. Which one will hit the ground first?

Accepted Answer

They both will hit at the same time.

Question 9

In the previous problem, which ball has the greater velocity at ground level?

Accepted Answer

The 'fired' ball

Question 10

A golfer chips a ball over a $$3.00\ \mathrm{m}$$ high tree that is $$14.0\ \mathrm{m}$$ away. The ball lands at the same level from which it was struck after traveling $$17.8\ \mathrm{m}$$ horizontally in $$2.24\ \mathrm{s}. $$What was the initial speed of the ball?

Accepted Answer

$$v_0$$ = \frac{17.8}{2.24 \cos 54.0^\circ} = 14.7\ \mathrm{m/s}$$

Question 11

Which of the following best describes the trajectory of a projectile launched horizontally from a height $$h$$?

Accepted Answer

A parabola described by $$y = h - \frac{g}{2v_0^2$} x^2$

Question 12

A small cart is rolling at constant velocity on a flat track and fires a ball straight up into the air as it moves. What happens to the ball after it is fired?

Accepted Answer

It falls right back into the cart.

Question 13

For a projectile launched at an angle $$	heta$$ with initial speed $$v_0$$, what is the time to reach the maximum height?

Accepted Answer

$$t_{max} = \frac{v_0$ \sin 	heta}{g}$$

Question 14

A projectile is launched from the ground with $$v_0 = 25$$\ \mathrm{m/s}$$ at $$	heta = 45^\circ$$. What is the maximum height reached by the projectile?

Accepted Answer

$$h_{max} = \frac{(v_0$ \sin \$$theta)^2$$}{2g} = \frac{(25 \sin 45^\$$circ)^2$$}{2 	imes 9.81} = 15.9\ \mathrm{m}$$

Question 15

Which launch angle gives the maximum range for a projectile (assuming the same initial and final elevation)?

Accepted Answer

$$45^\circ$$

Question 16

A projectile is launched with $$v_0 = 10$$\ \mathrm{m/s}$$ at $$	heta = 60^\circ$$. What is the horizontal component of its velocity?

Accepted Answer

$$v_{0x} = v_0$ \cos 	heta = 10 \cos 60^\circ = 5.0\ \mathrm{m/s}$$

Question 17

Which of the following statements about projectile motion (ignoring air resistance) is correct?

Accepted Answer

The horizontal and vertical motions are independent of each other.

Study Guide: Two-Dimensional Kinematics and Projectile Motion (Physics with Calculus)

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