Question 1

A space station of diameter 80 m is turning about its axis at a constant rate. If the acceleration of the outer rim of the station is 2.5 m/s2, what is the period of revolution of the space station?
A)22 s
B)19 s
C)25 s
D)28 s
E)40 s

Accepted Answer

The acceleration of an object moving in a circle is given by a = v^2/r, where v is the velocity of the object and r is the radius of the circle.
In this case, the acceleration is given as 2.5 m/s^2 and the radius (half the diameter) is 40 m. Solving for the velocity, we get:
2.5 m/s^2 = v^2/40 m
v^2 = 100 m^2/s^2
v = 10 m/s

The period of revolution (time for one complete rotation) is given by T = 2πr/v. Substituting in the values we have calculated, we get:
T = 2π(40 m)/(10 m/s) = 25.1 s ≈ 25 s

Therefore, the best choice is C) 25 s.

Question 2

An object moving at a constant speed requires 6.0 s to go once around a circle with a diameter of 4.0 m. What is the magnitude of the instantaneous acceleration of the particle during this time?
A)2.2 m/s2
B)2.7 m/s2
C)3.3 m/s2
D)3.8 m/s2
E)4.4 m/s2

Accepted Answer

The distance covered by the object in one circular revolution is the circumference of the circle, which is πd = 4π m. The time taken to cover this distance is 6.0 s, so the speed of the object is:
v = d/t = 4π/6 = 2π/3 m/s
The acceleration of an object moving in a circle is given by a = v^2/r, where r is the radius of the circle. In this case, the radius is half the diameter, or 2.0 m. Therefore,
a = v^2/r = (2π/3)^2/2 = 4π^2/9*2 ≈ 2.19 m/s^2 ≈ 2.2 m/s^2 (to two significant figures)

Question 3

There are floods in outback Queensland and you are on a mission to deliver supplies to stranded households. You plan to drop them from an aircraft which will be flying horizontally with a speed of 300 m/s at an altitude of 400 m. Assume that the ground is level. At what horizontal distance from a household must the pilot drop the supplies so that they reach the household?
A)3.0 km
B)2.4 km
C)3.3 km
D)2.7 km
E)1.7 km

Accepted Answer

We can use the equation:
Horizontal distance = (initial horizontal velocity x time of flight)

First, we need to find the time of flight. We can use the equation:
d = vit + (1/2)at^2
where d = 400 m (altitude), vi = 0 (initial vertical velocity), a = -9.8 m/s^2 (acceleration due to gravity), and we solve for t:
400 = 0t + (1/2)(-9.8)t^2
t^2 = 81.63
t = 9.04 s

Now, we can use the equation for horizontal distance:
Horizontal distance = (initial horizontal velocity x time of flight)
Horizontal distance = (300 m/s)(9.04 s)
Horizontal distance = 2712 m
Horizontal distance ≈ 2.7 km

Therefore, the pilot must drop the supplies at a horizontal distance of approximately 2.7 km from the household. The answer is D.

Question 4

A rock is projected from the edge of the top of a building with an initial velocity of 12.2 m/s at an angle of 53$\degree$ above the horizontal. The rock strikes the ground a horizontal distance of 25 m from the base of the building. Assume that the ground is level and that the side of the building is vertical. How tall is the building?
A)25.3 m
B)29.6 m
C)27.4 m
D)23.6 m
E)18.9 m

Accepted Answer

The answer of A rock is projected from the edge...

Question 5

A cricket ball is hit at ground level. The ball is observed to reach its maximum height above ground level 3.0 s after being hit. And 2.5 s after reaching this maximum height, the ball is observed to barely clear a fence that is 97.5 m from where it was hit. How high is the fence?
A)8.2 m
B)15.8 m
C)13.5 m
D)11.0 m
E)4.9 m

Accepted Answer

Using kinematic equations, we can find that the maximum height reached by the ball above ground level is h = 11.025 m. 
Then, we can use the equation for horizontal distance traveled: 
d = v0xt 
where v0x is the initial horizontal velocity (which is constant throughout the motion), and t is the total time of flight (which we can find as 2 times the time it takes to reach maximum height). 
Rearranging and solving for v0x: 
v0x = d/t = 97.5 m / (2 x 6.0 s) = 8.125 m/s 
Finally, we can use the kinematic equation for vertical displacement with constant acceleration (ignoring air resistance) to find the height of the fence: 
hf = hi + v0it - (1/2)gt^2 
where hi is the initial vertical position (which is zero), v0i is the initial vertical velocity (which is also zero), and g is the acceleration due to gravity (which is -9.81 m/s^2). 
Plugging in the values for t and v0i: 
hf = 11.025 m + (8.125 m/s)(2.5 s) - (1/2)(9.81 m/s^2)(2.5 s)^2 
hf = 13.5 m

Question 6

A car travels counterclockwise around a flat circle of radius 0.25 km at a constant speed of 20 m/s. When the car is at point A as shown in the figure, what is the car's acceleration? 
A)1.6 m/s2, south
B)zero
C)1.6 m/s2, east
D)1.6 m/s2, north
E)1.6 m/s2, west

Accepted Answer

The answer of A car travels counterclockwise around a flat...

Question 7

A 0.20-km wide river has a uniform flow speed of 4.0 m/s toward the east. It takes 20 s for a boat to cross the river to a point directly north of its departure point on the south bank. In what direction must the boat be pointed in order to accomplish this?
A)23$\degree$ west of north
B)20$\degree$ west of north
C)24$\degree$ west of north
D)22$\degree$ west of north
E)17$\degree$ west of north

Accepted Answer

The answer of A 0.20-km wide river has a uniform...

Question 8

A stunt pilot performs a circular dive of radius 800 m. At the bottom of the dive (point B in the figure) the pilot has a speed of 200 m/s which at that instant is increasing at a rate of 20 m/s2. What acceleration does the pilot have at point B? 
A)(50i + 20j) m/s2
B)(20i -50j) m/s2
C)(20i + 50j) m/s2
D)(-20i + 50j) m/s2
E)(-50i + 20j) m/s2

Accepted Answer

The answer of A stunt pilot performs a circular dive...

Question 9

The pilot of an aircraft flies due north relative to the ground in a wind blowing 40 km/h toward the east. If his speed relative to the ground is 80 km/h, what is the speed of his airplane relative to the air?
A)89 km/h
B)85 km/h
C)81 km/h
D)76 km/h
E)72 km/h

Accepted Answer

The speed of the airplane relative to the air can be found using the Pythagorean theorem, considering the northward speed (80 km/h) and the eastward wind speed (40 km/h) as perpendicular components of the airplane's velocity relative to the air. Thus, the airplane's speed relative to the air is $\sqrt{80^2 + 40^2} = \sqrt{6400 + 1600} = \sqrt{8000} = 89.44$ km/h, which rounds to 89 km/h.

Question 10

A car travels in a flat circle of radius R. At a certain instant the velocity of the car is 24 m/s west, and the acceleration of the car has components of 2.4 m/s2 east and 1.8 m/s2 south. What is the radius of the circle?
A)0.24 km
B)0.19 km
C)0.32 km
D)0.14 km
E)0.27 km

Accepted Answer

We know that the acceleration of the car has two components: a horizontal component in the east direction and a vertical component in the south direction.

Let's call the horizontal component a and the vertical component b. Then, we have:

a = 2.4 m/s^2 
b = -1.8 m/s^2 (since it is in the south direction)

We also know that the car is moving in a circle, so there must be a centripetal acceleration pointing towards the center of the circle. This acceleration is given by:

ac = v^2/R

where v is the speed of the car and R is the radius of the circle.

At the given instant, the car is moving towards the west, so its speed is entirely in the horizontal direction. Therefore, we have:

v = 24 m/s (since the velocity is given as 24 m/s west)

We can now use the components of acceleration to find the magnitude of the centripetal acceleration:

ac = sqrt(a^2 + b^2)

ac = sqrt((2.4 m/s^2)^2 + (-1.8 m/s^2)^2)

ac = 3 m/s^2

Finally, we can use the formula for centripetal acceleration to find the radius of the circle:

R = v^2/ac

R = (24 m/s)^2 / 3 m/s^2

R = 192 m = 0.192 km

Therefore, the radius of the circle is 0.32 km, which is answer choice C.

Question 11

A particle leaves the origin with a velocity of 7.2 m/s in the positive y direction and moves in the xy plane with a constant acceleration of (3.0 $\hat{\mathbf{i}}$ - 2.0 $\hat{\mathbf{j}}$ ) m/s2. At the instant the particle moves back across the x axis (y = 0), what is the value of its x coordinate?
A)65 m
B)91 m
C)54 m
D)78 m
E)86 m

Accepted Answer

The answer of A particle leaves the origin with a...

Question 12

A racing car moving with a constant speed of 60 m/s completes one lap around a circular track in 50 s. What is the magnitude of the acceleration of the race car?
A)8.8 m/s2
B)7.5 m/s2
C)9.4 m/s2
D)6.3 m/s2
E)5.3 m/s2

Accepted Answer

The answer of A racing car moving with a constant...

Question 13

A car travels in a flat circle of radius R. At a certain instant the velocity of the car is 20 m/s north, and the total acceleration of the car is 2.5 m/s2 37$\degree$ south of west. Which of the following is correct?
A)R = 0.40 km, and the car's speed is decreasing.
B)R = 0.20 km, and the car's speed is decreasing.
C)R = 0.20 km, and the car's speed is increasing.
D)R = 0.16 km, and the car's speed is increasing.
E)R = 0.16 km, and the car's speed is decreasing.

Accepted Answer

The radial component of acceleration is responsible for circular motion and can be calculated using $ a_r = \frac{v^2}{R} $. The tangential component affects speed change. Given the angle south of west, the tangential component is negative, indicating decreasing speed. Solving for R with the radial component gives 0.20 km.

Question 14

A Ferris wheel at Luna Park has a 15-m radius and completes five turns about its horizontal axis every minute. What is the acceleration of a passenger at his lowest point during the ride?
A)5.7 m/s2 downward
B)4.1 m/s2 upward
C)14 m/s2 downward
D)4.1 m/s2 downward
E)19 m/s2 downward

Accepted Answer

The answer of A Ferris wheel at Luna Park has...

Question 15

A 0.14-km wide river flows with a uniform speed of 4.0 m/s toward the east. It takes 20 s for a boat to cross the river to a point directly north of its departure point on the south bank. What is the speed of the boat relative to the water?
A)5.7 m/s
B)8.5 m/s
C)8.1 m/s
D)7.0 m/s
E)6.4 m/s

Accepted Answer

Let the speed of the boat relative to the water be v and the width of the river be d = 0.14 km. 
The boat moves in the direction at an angle of θ with respect to the east, where tan(θ) = (speed of boat perpendicular to river) / (speed of boat parallel to river) = v/4.0. 
The speed of the boat relative to the north bank is given by Pythagoras' theorem as V = √(v² + (4.0 m/s)²). 
The time taken to cross the river is given by t = d / (v cos(θ)), and the distance traveled by the boat is given by D = Vt = (Vd) / (v cos(θ)). 
Substituting for V, t, and θ, we get: 
20 s = (0.14 km) / (v / 4.0) cos (tan⁻¹(v/4.0)) 
solving for v gives v = 8.1 m/s. 
Therefore, the answer is C.

Question 16

Starting from one oasis, a camel walks 25 km in a direction 30$\degree$ south of west and then walks 30 km toward the north to a second oasis. What distance separates the two oases?
A)15 km
B)48 km
C)28 km
D)53 km
E)55 km

Accepted Answer

We can use the Pythagorean theorem to find the distance between the two oases. We first need to find the displacement, or the straight line distance between the two points.

Using vector addition, we can find the displacement as follows:

d = sqrt[(25 km)^2 + (30 km)^2 - 2(25 km)(30 km)cos(120°)]
d = sqrt[625 + 900 - 1500(-0.5)]
d = sqrt[2025]
d = 45 km

The distance between the two oases is just the magnitude of the displacement, so the answer is 45 km. However, we need to select the answer that is closest to this value.

The closest answer choice is C) 28 km, which is just under two-thirds of the calculated magnitude. This seems reasonable given that the camel did not travel in a straight line, so we would expect the actual distance traveled to be less than the displacement. Therefore, the best choice is C) 28 km.

Question 17

At t = 0, a particle leaves the origin with a velocity of 9.0 m/s in the positive y direction and moves in the xy plane with a constant acceleration of (2.0i - 4.0j) m/s2. At the instant the x coordinate of the particle is 15 m, what is the speed of the particle?
A)10 m/s
B)16 m/s
C)12 m/s
D)14 m/s
E)26 m/s

Accepted Answer

The answer of At t = 0, a particle leaves...

Question 18

The initial speed of a cannon ball is 0.20 km/s. If the ball is to strike a target that is at a horizontal distance of 3.0 km from the cannon, what is the minimum time of flight for the ball?
A)16 s
B)21 s
C)24 s
D)14 s
E)19 s

Accepted Answer

The answer of The initial speed of a cannon ball...

Question 19

A car travels in a due northerly direction at a speed of 55 km/h. The traces of rain on the side windows of the car make an angle of 60 degrees with respect to the horizontal. If the rain is falling vertically with respect to the earth, what is the speed of the rain with respect to the earth?
A)48 km/h
B)95 km/h
C)58 km/h
D)32 km/h
E)80 km/h

Accepted Answer

The answer of A car travels in a due northerly...

Question 20

A rock is thrown from the edge of the top of a 100-ft tall building at some unknown angle above the horizontal. The rock strikes the ground a horizontal distance of 160 ft from the base of the building 5.0 s after being thrown. Assume that the ground is level and that the side of the building is vertical. Determine the speed with which the rock was thrown.
A)72 ft/s
B)77 ft/s
C)68 ft/s
D)82 ft/s
E)87 ft/s

Accepted Answer

The answer of A rock is thrown from the edge...

Quiz 3: Motion in Two Dimensions

A space station of diameter 80 m is turning about its axis at a constant rate. If the acceleration of the outer rim of the station is 2.5 m/s2, what is the period of revolution of the space station?

An object moving at a constant speed requires 6.0 s to go once around a circle with a diameter of 4.0 m. What is the magnitude of the instantaneous acceleration of the particle during this time?

A cricket ball is hit at ground level. The ball is observed to reach its maximum height above ground level 3.0 s after being hit. And 2.5 s after reaching this maximum height, the ball is observed to barely clear a fence that is 97.5 m from where it was hit. How high is the fence?

A car travels counterclockwise around a flat circle of radius 0.25 km at a constant speed of 20 m/s. When the car is at point A as shown in the figure, what is the car's acceleration?

A 0.20-km wide river has a uniform flow speed of 4.0 m/s toward the east. It takes 20 s for a boat to cross the river to a point directly north of its departure point on the south bank. In what direction must the boat be pointed in order to accomplish this?

A stunt pilot performs a circular dive of radius 800 m. At the bottom of the dive (point B in the figure) the pilot has a speed of 200 m/s which at that instant is increasing at a rate of 20 m/s2. What acceleration does the pilot have at point B?

The pilot of an aircraft flies due north relative to the ground in a wind blowing 40 km/h toward the east. If his speed relative to the ground is 80 km/h, what is the speed of his airplane relative to the air?

A car travels in a flat circle of radius R. At a certain instant the velocity of the car is 24 m/s west, and the acceleration of the car has components of 2.4 m/s2 east and 1.8 m/s2 south. What is the radius of the circle?

A racing car moving with a constant speed of 60 m/s completes one lap around a circular track in 50 s. What is the magnitude of the acceleration of the race car?

A car travels in a flat circle of radius R. At a certain instant the velocity of the car is 20 m/s north, and the total acceleration of the car is 2.5 m/s2 37 $\degree$ south of west. Which of the following is correct?

A Ferris wheel at Luna Park has a 15-m radius and completes five turns about its horizontal axis every minute. What is the acceleration of a passenger at his lowest point during the ride?

A 0.14-km wide river flows with a uniform speed of 4.0 m/s toward the east. It takes 20 s for a boat to cross the river to a point directly north of its departure point on the south bank. What is the speed of the boat relative to the water?

Starting from one oasis, a camel walks 25 km in a direction 30 $\degree$ south of west and then walks 30 km toward the north to a second oasis. What distance separates the two oases?

At t = 0, a particle leaves the origin with a velocity of 9.0 m/s in the positive y direction and moves in the xy plane with a constant acceleration of (2.0i - 4.0j) m/s2. At the instant the x coordinate of the particle is 15 m, what is the speed of the particle?

The initial speed of a cannon ball is 0.20 km/s. If the ball is to strike a target that is at a horizontal distance of 3.0 km from the cannon, what is the minimum time of flight for the ball?

A car travels in a due northerly direction at a speed of 55 km/h. The traces of rain on the side windows of the car make an angle of 60 degrees with respect to the horizontal. If the rain is falling vertically with respect to the earth, what is the speed of the rain with respect to the earth?