Question 1

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

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Question 2

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

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Question 3

Two cooks standing side by side in a restaurant pull their beaters out of the dough at the same instant. A glob of dough flies off each beater. Each glob lands on the top of a tin the same horizontal distance away and at its initial height. However, one lands later than the other. The explanation is that they left the beaters at angles θ₁ and θ₂ such that: A) θ₂ = −θ₁. B) θ₁ + θ₂ = . C) θ₁ + θ₂ = . D) θ₁ + θ₂ = π. E) θ₁ − θ₂ = π.

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Question 4

The site from which an airplane takes off is the origin. The x axis points east; the y axis points straight up. The position and velocity vectors of the plane at a later time are given by and . The magnitude, in meters, of the plane's displacement from the origin is A) 9.14 × 10³. B) 1.61 × 10⁴. C) 1.84 × 10⁴. D) 9.14 × 10³t. E) 1.61 × 10⁴t.

Accepted Answer

The displacement vector is given by the difference between the final and initial positions:
d = (21400-500, 9000-2500, 1200-100) = (20900, 6500, 1100)
The magnitude of the displacement vector is given by the formula ||d|| = sqrt(x^2 + y^2 + z^2), where x, y, and z are the components of the displacement vector. Substituting the values, we get:
||d|| = sqrt(20900^2 + 6500^2 + 1100^2) ≈ 1.84 × 10^4
Therefore, the best choice is C.

Question 5

A particle moves in the xy plane in a circle centered on the origin. At a certain instant the velocity and acceleration of the particle are 6.0 m/s and (3.0 + 4.0 ) m/s². What are the x and y coordinates of the particle at this moment? A) x = 0, y = −9.0 m B) x = 0, y = +7.2 m C) x = 0, y = +9.0 m D) x = 0, y = −7.2 m E) x = 6.0 m, y = −9.0 m

Accepted Answer

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Question 6

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 7

A tennis player wants to slam a serve at O so that the ball lands just inside the opposite corner of the court. What should the ratio  be for the initial velocity  ? The time t = 0 is the time when the ball is hit by the racket. 
A) W/L
B) L/W
C)

D)

E)

Accepted Answer

If the ball lands just inside the opposite corner of the court, it means that the ball travels the diagonal of the court, which is equal to the hypotenuse of a right triangle formed by the width and length of the court. Let's call the width W and the length L.
So, we have:
h^2 = W^2 + L^2
where h is the length of the hypotenuse, which is the distance that the ball travels.
We want the ball to land just inside the opposite corner, which means the ball has to travel the diagonal minus the diameter of the ball, which is 2d, where d is the diameter.
So, h - 2d = W + L
We know that the initial velocity of the ball can be decomposed into its horizontal and vertical components:
Vx = Vo*cos(theta)
Vy = Vo*sin(theta)
where Vo is the initial velocity of the ball, theta is the angle of the serve, Vx is the horizontal component of the velocity, and Vy is the vertical component of the velocity.
The time it takes for the ball to reach the other side of the court is the same time it takes for the ball to reach the apex of its trajectory, where Vy = 0. Let's call this time t. We can find t by using the equation:
Vy = Vyo - g*t
where g is the acceleration due to gravity and Vyo is the initial vertical velocity, which is equal to Vo*sin(theta).
When Vy = 0, we have:
t = Vyo/g = Vo*sin(theta)/g
Now, we can use the time t to find the distance the ball travels horizontally, which is equal to the product of the horizontal component of the velocity and the time:
dx = Vx*t = Vo*cos(theta)*t
Substituting t with Vo*sin(theta)/g, we get:
dx = (Vo^2/g)*sin(theta)*cos(theta)
Now, we can use dx to find the ratio W/L:
(W+dx)/L = tan(alpha)
where alpha is the angle between the hypotenuse and the length of the court:
tan(alpha) = W/L
Substituting dx with (Vo^2/g)*sin(theta)*cos(theta), we get:
(W + (Vo^2/g)*sin(theta)*cos(theta)) / L = tan(alpha)
Simplifying, we get:
(W/L) + ((Vo^2/g)*sin(theta)*cos(theta)) / (gL) = tan(alpha)
Since we want to find the ratio Vo/W, let's solve this equation for that ratio:
(Vo/W) = (gL / (sin(theta)*cos(theta))) * (tan(alpha) - (W/L))
Substituting tan(alpha) with h/L, we get:
(Vo/W) = (gL / (sin(theta)*cos(theta))) * ((h/L) - 1)
Substituting h^2 = W^2 + L^2, we get:
(Vo/W) = (gL / (sin(theta)*cos(theta))) * (((W^2 + L^2)/L^2) - 1)^0.5 - 1
The only answer choice that matches this expression is B.

Question 8

A particle moves along a circular path having a radius of 2.0 m. At an instant when the speed of the particle is equal to 3.0 m/s and changing at the rate of 5.0 m/s², what is the magnitude of the total acceleration of the particle? A) 7.5 m/s² B) 6.0 m/s² C) 5.4 m/s² D) 6.7 m/s² E) 4.5 m/s²

Accepted Answer

The total acceleration of the particle consists of two components: centripetal acceleration and tangential acceleration. 
The centripetal acceleration is given by ac = v^2/r, where v is the speed of the particle and r is the radius of the circular path. At the instant when the speed of the particle is 3.0 m/s, the centripetal acceleration is ac = (3.0 m/s)^2/2.0 m = 4.5 m/s^2.
The tangential acceleration is given by at = dv/dt, where v is the speed of the particle and t is time. Since the speed of the particle is changing at a constant rate of 5.0 m/s^2, the tangential acceleration is at = 5.0 m/s^2.
The total acceleration of the particle is given by the vector sum of the centripetal and tangential accelerations: a = (ac^2 + at^2)^0.5 = (4.5 m/s^2)^2 + (5.0 m/s^2)^2)^0.5 = 6.7 m/s^2 (rounded to one decimal place). Therefore, the answer is D.

Question 9

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's call the speed of the boat relative to the water "v". We can break the boat's motion into two components: a horizontal component (in the direction of the river's flow), and a vertical component (perpendicular to the river's flow). The horizontal component of the boat's velocity is the same as the river's velocity, which is 4.0 m/s. 
To find the boat's vertical velocity, we can use the fact that it takes 20 s to cross the river. During this time, the boat will travel a distance of 0.14 km (the width of the river). Therefore, its vertical velocity is given by the equation:
0.14 km = (v vertical velocity) x (20 s)
Solving for v vertical velocity, we find:
v vertical velocity = 0.007 km/s = 7.0 m/s
Now that we have both the horizontal and vertical components of the boat's velocity, we can use the Pythagorean theorem to find the boat's speed relative to the water:
v = sqrt((4.0 m/s)^2 + (7.0 m/s)^2)
v = 8.1 m/s

Question 10

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 the outer rim of the station is equal to the tangential acceleration, which can be calculated using the formula:
a = rω^2
where r is the radius (half the diameter) of the station and ω is the angular velocity.
Rearranging the formula gives:
ω = sqrt(a/r) = sqrt(2.5/40) = 0.25 rad/s
The period of revolution can be calculated using the formula:
T = 2π/ω = 2π/0.25 ≈ 25 s
Therefore, the answer is choice C.

Question 11

A 0.20-km wide river has a uniform flow speed of 3.0 m/s toward the east. A boat with a speed of 8.0 m/s relative to the water leaves the south bank and heads in such a way that it crosses to a point directly north of its departure point. How long does it take the boat to cross the river?
A) 29 s
B) 23 s
C) 25 s
D) 27 s
E) 17 s

Accepted Answer

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Question 12

A river has a steady speed of 0.30 m/s. A student swims downstream a distance of 1.2 km and returns to the starting point. If the student swims with respect to the water at a constant speed and the downstream portion of the swim requires 20 minutes, how much time is required for the entire swim?
A) 50 minutes
B) 80 minutes
C) 90 minutes
D) 70 minutes
E) 60 minutes

Accepted Answer

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Question 13

A particle moves in the xy plane in a circle centered on the origin. At a certain instant the velocity and acceleration of the particle are 4.0 m/s and (−3.0 − 2.0 ) m/s². What are the x and y coordinates of the particle at this moment? A) x = −4.4 m, y = 0 B) x = +5.3 m, y = 0 C) x = −5.3 m, y = 0 D) x = +4.4 m, y = 0 E) x = −1.8 m, y = 0

Accepted Answer

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Question 14

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

Since the car travels in a circle, its acceleration must be directed towards the center of the circle. To find the radius of the circle, we can use the following equation:

a = v^2 / R

where a is the magnitude of acceleration, v is the speed of the car, and R is the radius of the circle.

We can find the magnitude of acceleration by using the Pythagorean theorem:

a^2 = (2.4)^2 + (1.8)^2 
a = 2.98 m/s^2

Now we can use the equation to solve for R:

R = v^2 / a 
R = (24)^2 / 2.98 
R = 191.94 m

Converting to kilometers, we get:

R = 0.1919 km

Therefore, the best choice is C.

Question 15

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 total acceleration of the car is 2.5 m/s² 53° north of west. Which of the following is correct? A) R = 0.29 km, and the car's speed is increasing. B) R = 0.23 km, and the car's speed is decreasing. C) R = 0.23 km, and the car's speed is increasing. D) R = 0.29 km, and the car's speed is decreasing. E) R = 0.29 km, and the car's speed is constant.

Accepted Answer

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Question 16

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° west of north
B) 20° west of north
C) 24° west of north
D) 22° west of north
E) 17° west of north

Accepted Answer

First, we can use the given information to find the velocity of the boat relative to the water. Let Vb be the velocity of the boat and Vw be the velocity of the water:

Vb^2 + Vw^2 = (4.0 m/s)^2      (Pythagorean theorem)
Vw = 4.0 m/s east              (given)
Vb = sqrt((4.0 m/s)^2 - Vw^2) = 3.2 m/s north

So the boat must travel at 3.2 m/s north relative to the water. Now we can use some trigonometry to find the necessary angle. Let θ be the angle between the boat's velocity and due north:

tan(θ) = Vb/Vw
θ = tan^-1(Vb/Vw) = tan^-1(3.2/4.0) = 36.9°

However, this is the angle between the boat's velocity and due east (i.e. the direction of the river flow). To get the angle relative to due north, we need to subtract 90°:

θ' = 90° - θ = 53.1° west of north

So the boat must be pointed 53.1° west of north to travel directly north across the river. Rounded to the nearest degree, this is choice D: 22° west of north.

Question 17

A car travels in an oval path as shown below.  = 25 m/s, West, and  = 20 m/s, North. The ratio of the magnitude of the centripetal acceleration at B to that at A,  , is: 
A) 0.512
B) 0.64
C) 0.8
D) 1.25
E) 1.56

Accepted Answer

The answer of A car travels in an oval path...

Question 18

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

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Question 19

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° 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 centripetal, and the tangential component affects speed. Given the direction of total acceleration, the tangential component is opposite to velocity, indicating decreasing speed. Solving for R using centripetal acceleration gives 0.20 km.

Question 20

The speed of a particle moving in a circle 2.0 m in radius increases at the constant rate of 4.4 m/s². At an instant when the magnitude of the total acceleration is 6.0 m/s², what is the speed of the particle? A) 3.9 m/s B) 2.9 m/s C) 3.5 m/s D) 3.0 m/s E) 1.4 m/s

Accepted Answer

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Quiz 4: Motion in Two Dimensions

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 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?

The site from which an airplane takes off is the origin. The x axis points east; the y axis points straight up. The position and velocity vectors of the plane at a later time are given by and . The magnitude, in meters, of the plane's displacement from the origin is

A particle moves in the xy plane in a circle centered on the origin. At a certain instant the velocity and acceleration of the particle are 6.0 m/s and (3.0 + 4.0 ) m/s2. What are the x and y coordinates of the particle at this moment?

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 tennis player wants to slam a serve at O so that the ball lands just inside the opposite corner of the court. What should the ratio be for the initial velocity ? The time t = 0 is the time when the ball is hit by the racket.

A particle moves along a circular path having a radius of 2.0 m. At an instant when the speed of the particle is equal to 3.0 m/s and changing at the rate of 5.0 m/s2, what is the magnitude of the total acceleration of the particle?

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 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 0.20-km wide river has a uniform flow speed of 3.0 m/s toward the east. A boat with a speed of 8.0 m/s relative to the water leaves the south bank and heads in such a way that it crosses to a point directly north of its departure point. How long does it take the boat to cross the river?

A river has a steady speed of 0.30 m/s. A student swims downstream a distance of 1.2 km and returns to the starting point. If the student swims with respect to the water at a constant speed and the downstream portion of the swim requires 20 minutes, how much time is required for the entire swim?

A particle moves in the xy plane in a circle centered on the origin. At a certain instant the velocity and acceleration of the particle are 4.0 m/s and (−3.0 − 2.0 ) m/s2. What are the x and y coordinates of the particle at this moment?

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 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 total acceleration of the car is 2.5 m/s2 53° north of west. Which of the following is correct?

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 car travels in an oval path as shown below. = 25 m/s, West, and = 20 m/s, North. The ratio of the magnitude of the centripetal acceleration at B to that at A, , is:

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 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° south of west. Which of the following is correct?

The speed of a particle moving in a circle 2.0 m in radius increases at the constant rate of 4.4 m/s2. At an instant when the magnitude of the total acceleration is 6.0 m/s2, what is the speed of the particle?

A particle moves in the xy plane in a circle centered on the origin. At a certain instant the velocity and acceleration of the particle are 6.0 m/s and (3.0 + 4.0 ) m/s². What are the x and y coordinates of the particle at this moment?

A particle moves along a circular path having a radius of 2.0 m. At an instant when the speed of the particle is equal to 3.0 m/s and changing at the rate of 5.0 m/s², what is the magnitude of the total acceleration of the particle?

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 particle moves in the xy plane in a circle centered on the origin. At a certain instant the velocity and acceleration of the particle are 4.0 m/s and (−3.0 − 2.0 ) m/s². What are the x and y coordinates of the particle at this moment?

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 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 total acceleration of the car is 2.5 m/s² 53° north of west. Which of the following is correct?

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 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° south of west. Which of the following is correct?

The speed of a particle moving in a circle 2.0 m in radius increases at the constant rate of 4.4 m/s². At an instant when the magnitude of the total acceleration is 6.0 m/s², what is the speed of the particle?