Q: Firefighters are using a hose to shoot water at a burning building. The water leaves the hose at a speed of $$25.0\ \mathrm{m/s}$$ and reaches a building $$45.0\ \mathrm{m}$$ away in $$3.00\ \mathrm{s}. $$Assuming projectile motion and neglecting air resistance, what is the angle of elevation $$\alpha of $$the hose above the horizontal?

$$\alpha = \arccos\left(\dfrac{45.0}{25.0 \times 3.00}\right)$$,\ \text{but using the horizontal velocity:}\ \alpha = \arccos\left(\dfrac{45.0}{25.0 \times 3.00}\right) \approx 69.5^\circ$

Q: A 124-kg balloon carrying a 22-kg basket is descending at a constant velocity of $$20.0\ \mathrm{m/s}$$ downward. A $$1.0\ \mathrm{kg}$$ stone is thrown horizontally from the basket at $$15.0\ \mathrm{m/s}$$ relative to the basket. If the stone hits the ground $$6.00\ \mathrm{s}$$ later, how high was the balloon when the stone was thrown?

$$h = v_{0y} t + \dfrac{1}{2} g t^2$ = 20.0 \times 6.00 + 0.5 \times 9.8 \times (6.00)^2$ = 120 + 176.4 = 296.4\ \mathrm{m}$$

Q: At the highest point in its trajectory, what is the acceleration of the water stream shot from the fire hose (ignoring air resistance)?

$$a = 9.8\ \mathrm{m/s^2$}$$\ \text{downward}

Q: A $$1.0\ \mathrm{kg}$$ stone is thrown from a balloon and hits the ground $$6.00\ \mathrm{s}$$ later. What is the vertical component of the stone's velocity just before it hits the ground (relative to the ground)?

$$v_y = v_{0y} + g t = -20.0 + 9.8 \times 6.00 = 38.8\ \mathrm{m/s}$$\ \text{downward}

Q: For the oscillating air-track glider, what is the total mechanical energy of the system if the amplitude is $$0.10\ \mathrm{m}$$ and the spring constant is $$k = 10.0\ \mathrm{N/m}$$?

$$E = \dfrac{1}{2} k A^2$ = 0.5 \times 10.0 \times (0.10)^2$ = 0.05\ \mathrm{J}$$

Q: After the stone is thrown from the balloon, how far horizontally from the basket is the stone when it hits the ground (relative to the ground)?

$$d = v_x t = 15.0 \times 6.00 = 90.0\ \mathrm{m}$$

Question 1

Firefighters are using a hose to shoot water at a burning building. The water leaves the hose at a speed of $$25.0\ \mathrm{m/s}$$ and reaches a building $$45.0\ \mathrm{m}$$ away in $$3.00\ \mathrm{s}. $$Assuming projectile motion and neglecting air resistance, what is the angle of elevation $$\alpha of $$the hose above the horizontal?

Accepted Answer

$$\alpha = \arccos\left(\dfrac{45.0}{25.0 	imes 3.00}ight)$$,\ 	ext{but using the horizontal velocity:}\ \alpha = \arccos\left(\dfrac{45.0}{25.0 	imes 3.00}ight) \approx 69.5^\circ$

Question 2

A 124-kg balloon carrying a 22-kg basket is descending at a constant velocity of $$20.0\ \mathrm{m/s}$$ downward. A $$1.0\ \mathrm{kg}$$ stone is thrown horizontally from the basket at $$15.0\ \mathrm{m/s}$$ relative to the basket. If the stone hits the ground $$6.00\ \mathrm{s}$$ later, how high was the balloon when the stone was thrown?

Accepted Answer

$$h = v_{0y} t + \dfrac{1}{2} g t^2$ = 20.0 	imes 6.00 + 0.5 	imes 9.8 	imes (6.00)^2$ = 120 + 176.4 = 296.4\ \mathrm{m}$$

Question 3

At the highest point in its trajectory, what is the acceleration of the water stream shot from the fire hose (ignoring air resistance)?

Accepted Answer

$$a = 9.8\ \mathrm{m/s^2$}$$\ 	ext{downward}

Question 4

A $$0.200\ \mathrm{kg}$$ air-track glider is attached to an ideal spring and set oscillating. If the maximum speed of the glider is $$0.50\ \mathrm{m/s}$$ and the amplitude is $$0.10\ \mathrm{m}$$, what is the angular frequency $$\omega of $$the oscillation?

Accepted Answer

$$\omega = \dfrac{v_{max}}{A} = \dfrac{0.50}{0.10} = 5.0\ \mathrm{rad/s}$$

Question 5

A stone is thrown horizontally from a descending balloon with a downward velocity of $$20.0\ \mathrm{m/s}. If $$the stone is thrown at $$15.0\ \mathrm{m/s}$$ relative to the basket, what is the stone's velocity relative to the ground immediately after being thrown?

Accepted Answer

$$\vec{v} = \sqrt{(15.0)^2$ + (20.0)^2$} = 25.0\ \mathrm{m/s}$$\ 	ext{at an angle below horizontal}

Question 6

If the water stream from the fire hose takes $$3.00\ \mathrm{s} to $$reach the building $$45.0\ \mathrm{m}$$ away, what is the horizontal component of the water's velocity?

Accepted Answer

$$v_x = \dfrac{45.0}{3.00} = 15.0\ \mathrm{m/s}$$

Question 7

A $$1.0\ \mathrm{kg}$$ stone is thrown from a balloon and hits the ground $$6.00\ \mathrm{s}$$ later. What is the vertical component of the stone's velocity just before it hits the ground (relative to the ground)?

Accepted Answer

$$v_y = v_{0y} + g t = -20.0 + 9.8 	imes 6.00 = 38.8\ \mathrm{m/s}$$\ 	ext{downward}

Question 8

For the oscillating air-track glider, what is the total mechanical energy of the system if the amplitude is $$0.10\ \mathrm{m}$$ and the spring constant is $$k = 10.0\ \mathrm{N/m}$$?

Accepted Answer

$$E = \dfrac{1}{2} k A^2$ = 0.5 	imes 10.0 	imes (0.10)^2$ = 0.05\ \mathrm{J}$$

Question 9

A fireman wants the water stream to hit the building at the highest possible point. Which angle of elevation should he choose (assuming the same initial speed and distance)?

Accepted Answer

The angle calculated from projectile motion equations for the given time and distance

Question 10

After the stone is thrown from the balloon, how far horizontally from the basket is the stone when it hits the ground (relative to the ground)?

Accepted Answer

$$d = v_x t = 15.0 	imes 6.00 = 90.0\ \mathrm{m}$$

Question 11

For the oscillating glider, if the period of oscillation is $$T = 1.26\ \mathrm{s}$$, what is the spring constant $$k$$?

Accepted Answer

$$k = \dfrac{4 \pi^2$ m}{T^2$} = \dfrac{4 	imes (3.14)^2$ 	imes 0.200}{(1.26)^2$} \approx 4.98\ \mathrm{N/m}$$

Question 12

A fireman wants to know how high above the ground the water will strike the building. Which equation should he use to find the vertical displacement $$y$$ after $$t = 3.00\ \mathrm{s}$$?

Accepted Answer

$$y = v_{0y} t - \dfrac{1}{2} g t^2$

Question 13

If the balloon continues descending at $$20.0\ \mathrm{m/s}$$, how high is the balloon when the stone hits the ground $$6.00\ \mathrm{s}$$ after being thrown?

Accepted Answer

$$h = h_0$ - v_{balloon} t = 296.4 - 20.0 	imes 6.00 = 176.4\ \mathrm{m}$$

Question 14

What is the speed of the water at the highest point in its trajectory after leaving the fire hose?

Accepted Answer

$$v = v_{0x}$$

Question 15

A glider attached to a spring oscillates with amplitude $$A$$ and angular frequency $$\omega. At $$what displacement from equilibrium is the kinetic energy equal to the potential energy?

Accepted Answer

$$x = \dfrac{A}{\sqrt{2}}$$

Question 16

A stone is thrown horizontally from a balloon. What is the stone's horizontal velocity relative to the basket just before it hits the ground?

Accepted Answer

$$v_x = 15.0\ \mathrm{m/s}$$

Question 17

If the air-track glider's mass is doubled but the spring constant remains the same, what happens to the period of oscillation?

Accepted Answer

The period increases by a factor of $$\sqrt{2}$$

Question 18

A fireman wants to increase the range of the water stream. Which of the following will increase the range, assuming the angle remains constant?

Accepted Answer

Increase the initial speed of the water

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