Question 1

Plane A is flying at 400 mph in the northeast direction relative to the earth.Plane B is flying at 500 mph in the north direction relative to the earth.What is the direction of motion of Plane B as observed from Plane A?
A)52.5 $\degree$ N of E
B)52.5 $\degree$ N or W
C)37.5 $\degree$ N of W
D)36.9 $\degree$N of W

Accepted Answer

The answer of Plane A is flying at 400 mph...

Question 2

A river flows due east at 3.0 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 10 m/s due north relative to the water.If no correction is made for the current,how far downstream does the boat move by the time it reaches the far shore?
A)6 m
B)30 m
C)60 m
D)90 m

Accepted Answer

The boat's motion can be broken down into two components: its velocity due north (10 m/s) and the river's velocity due east (3.0 m/s). The boat's actual velocity relative to the shore is the vector sum of these two velocities. Using the Pythagorean theorem, we can find the magnitude of this velocity to be `sqrt(10^2 + 3^2) = sqrt(109)`.

The angle between the boat's velocity and the river's velocity is given by `tan(theta) = 3/10`, which implies `theta = 17.98 degrees`.

The boat takes `t = 300/10 = 30 s` to cross the river. In this time, the boat will have traveled a horizontal distance equal to its velocity relative to the shore multiplied by the time, or `10 * 30 = 300 m`. However, due to the river's current, the boat will also have drifted downstream. The horizontal distance traveled downstream is given by `d = vt = 3 * 30 = 90 m`. Therefore, the boat moves `90 m` downstream by the time it reaches the far shore.

Question 3

Vectors  ,  ,and  have magnitudes 6,11,and 20.When these vectors are added,what is the least possible magnitude of their resultant?
A)25
B)15
C)2
D)3

Accepted Answer

The answer of Vectors  ,  ,and  have...

Question 4

Four vectors all have the same magnitude.Vector 1 is at 30 $\degree$,Vector 2 is at 135 $\degree$,vector 3 is at 240 $\degree$,and Vector 4 is at 315 $\degree$.Which vector has the greatest magnitude x-component and which vector has the greatest magnitude y-component?
A)Vector 1,Vector 2
B)Vector 3,Vector 4
C)Vector 1,Vector 3
D)Vector 3,Vector 2

Accepted Answer

Vector 1 has the greatest x-component because it is closest to the positive x-axis. Vector 3 has the greatest y-component because it is closest to the negative y-axis.

Question 5

A boat moves through the water in a river at a speed of 8 m/s relative to the water.The boat makes a trip downstream and then makes a return trip upstream to the original starting place.Which trip takes longer?
A)the downstream trip
B)the upstream trip
C)Both trips take the same amount of time.
D)The answer cannot be figured without knowing the speed of the river flow.

Accepted Answer

The upstream trip takes longer because the river's flow opposes the boat's motion, effectively reducing its speed relative to the ground.

Question 6

A particle moves east at constant velocity  for a time interval DT.It then moves north at a constant velocity,with the same speed as before,for another time interval DT.Finally it moves east again with the original velocity.At the instant an additional time interval DT has elapsed,which of the following are true about the average velocity and the average acceleration for the motion described?
A)The average velocity is  and the average acceleration is zero.
B)The average velocity is  and the average acceleration is not zero.
C)The average velocity is not  and the average acceleration is zero.
D)The average velocity is not  and the average acceleration is not zero.

Accepted Answer

The answer of A particle moves east at constant velocity...

Question 7

A projectile is fired at an angle of elevation of 60 $\degree$.Neglecting air resistance,what are possible angles in flight between the acceleration vector and the velocity vector?
A)160 $\degree$ and 40 $\degree$
B)20 $\degree$ and 70 $\degree$
C)90 $\degree$ and 60 $\degree$
D)none of the above

Accepted Answer

When a projectile is fired at an angle of elevation, the acceleration vector (due to gravity) is always straight down towards the ground. The only variable is the direction of the velocity vector.

At the top of the projectile's trajectory, the velocity vector is perpendicular to the acceleration vector, so the angle between them is 90 degrees. As the projectile begins to fall back towards the ground, the angle between the velocity and acceleration vectors decreases.

Since the angle of elevation in this problem is 60 degrees, the initial velocity vector is also 60 degrees above the horizontal. Therefore, at the top of the trajectory, the velocity vector is pointing straight up and the angle between the velocity and acceleration vectors is 90 degrees.

As the projectile continues to fall, the angle between the velocity and acceleration vectors decreases. However, the only answer choice that includes 90 degrees as a possibility is C. Therefore, C is the best choice.

Question 8

A boat moves at 10.0 m/s relative to the water.If the boat is in a river where the current is 2.00 m/s,how long does it take the boat to make a complete round trip of 1 000 m upstream followed by a 1 000-m trip downstream?
A)200 s
B)203 s
C)208 s
D)250 s

Accepted Answer

Let's first calculate the time taken to travel 1000m upstream:
Time taken = distance/speed relative to the water
Speed of boat upstream = 10 - 2 = 8 m/s
Time taken to travel 1000m upstream = 1000/8 = 125 s

Now, let's calculate the time taken to travel 1000m downstream:
Speed of boat downstream = 10 + 2 = 12 m/s
Time taken to travel 1000m downstream = 1000/12 = 83.33 s

Total time taken for round trip = time taken to go upstream + time taken to go downstream
= 125 s + 83.33 s
= 208.33 s

Rounding off to the nearest second, we get the answer as 208 s (option C).

Question 9

A plane is moving due north,directly towards its destination.Its airspeed is 200 mph.A constant breeze is blowing from west to east at 20 mph.In which direction is the plane pointed?
A)5.7 $\degree$ W of N
B)10 $\degree$ W of N
C)22 $\degree$W of N
D)11 $\degree$E of N

Accepted Answer

The plane is pointed 5.7° W of N. This is because the airspeed is 200 mph and the wind speed is 20 mph. The plane is moving directly north, so the wind is blowing from the west. This means that the plane is being pushed to the east by the wind. The angle between the plane's airspeed and the wind speed is 5.7°.

Question 10

A plane is moving due north,directly towards its destination.Its airspeed is 200 mph.A constant breeze is blowing from west to east at 40 mph.How long will it take for the plane to travel 200 miles north?
A)one hour
B)more than one hour
C)less than one hour
D)more information is needed

Accepted Answer

The plane's airspeed is 200 mph, meaning it moves 200 miles in 1 hour without considering the wind. However, the wind is blowing at 40 mph from west to east. This means that the plane will need to travel slightly longer than the expected hour to reach its destination, as it will be pushed eastward by the wind. Without knowing the angle between the plane's flight path and the wind direction it is impossible to calculate the exact time needed, but we know for sure that the answer is not A (less than one hour) or C (more than one hour). Therefore, the best choice is B (more than one hour).

Question 11

A boat travels upstream and after one hour has gone 10 km.The boat next travels downstream and after one hour has gone 14 km.If the boat's speed relative to the water is constant,what is the speed of the current in the river?
A)1 km/h
B)2 km/h
C)3 km/h
D)4 km/h

Accepted Answer

Let's assume that the speed of the boat in still water is B km/h and the speed of the current is C km/h.

When the boat travels upstream, it is moving against the current which means its effective speed is B - C km/h.

So, we can write: distance = speed x time
10 = (B - C) x 1

Similarly, when the boat travels downstream, it is moving with the current which means its effective speed is B + C km/h.

So, we can write: distance = speed x time
14 = (B + C) x 1

We now have two equations and two unknowns (B and C).

Solving these equations, we get:

B = 12 km/h
C = 2 km/h

Therefore, the speed of the current in the river is 2 km/h, which is option B.

Question 12

Vector 1 is 7 units long and is at 70 $\degree$.Vector 2 is 5 units long and is at 225 $\degree$.Vector 3 is 3 units long and is at 150 $\degree$.Which vector has equal magnitude components?
A)Vector 1
B)Vector 2
C)Vector 3
D)None of the vectors has equal magnitude components.

Accepted Answer

The answer of Vector 1 is 7 units long and...

Question 13

Plane A is flying at 400 mph in the northeast direction relative to the earth.Plane B is flying at 500 mph in the north direction relative to the earth.What is the speed of Plane B as observed from Plane A?
A)900 mph
B)640 mph
C)357 mph
D)100 mph

Accepted Answer

To find the speed of Plane B as observed from Plane A, we need to calculate the relative velocity. Plane A's velocity can be broken down into components: 400 mph northeast means 400/sqrt(2) mph north and 400/sqrt(2) mph east. Plane B is moving 500 mph north. The relative velocity in the north direction is 500 - 400/sqrt(2) and in the east direction is -400/sqrt(2). The magnitude of the relative velocity is sqrt((500 - 400/sqrt(2))^2 + (-400/sqrt(2))^2) which is approximately 357 mph.

Question 14

A plane is moving due north,directly towards its destination.Its airspeed is 200 mph.A constant breeze is blowing from west to east at 30 mph.At what rate is the plane moving north?
A)198 mph
B)193 mph
C)188 mph
D)180 mph

Accepted Answer

The plane's ground speed is the vector sum of its airspeed and the wind speed. The airspeed is 200 mph due north, and the wind speed is 30 mph due east. The vector sum of these two vectors is a vector with magnitude 200 mph and direction 19.3 degrees east of north. The rate at which the plane is moving north is the component of its ground speed in the north direction, which is 200 mph * cos(19.3 degrees) = 198 mph.

Quiz 3: Vectors and Two-Dimensional Motion

Plane A is flying at 400 mph in the northeast direction relative to the earth.Plane B is flying at 500 mph in the north direction relative to the earth.What is the direction of motion of Plane B as observed from Plane A?

A river flows due east at 3.0 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 10 m/s due north relative to the water.If no correction is made for the current,how far downstream does the boat move by the time it reaches the far shore?

Vectors , ,and have magnitudes 6,11,and 20.When these vectors are added,what is the least possible magnitude of their resultant?

Four vectors all have the same magnitude.Vector 1 is at 30 $\degree$ ,Vector 2 is at 135 $\degree$ ,vector 3 is at 240 $\degree$ ,and Vector 4 is at 315 $\degree$ .Which vector has the greatest magnitude x-component and which vector has the greatest magnitude y-component?

A boat moves through the water in a river at a speed of 8 m/s relative to the water.The boat makes a trip downstream and then makes a return trip upstream to the original starting place.Which trip takes longer?

A projectile is fired at an angle of elevation of 60 $\degree$ .Neglecting air resistance,what are possible angles in flight between the acceleration vector and the velocity vector?

A boat moves at 10.0 m/s relative to the water.If the boat is in a river where the current is 2.00 m/s,how long does it take the boat to make a complete round trip of 1 000 m upstream followed by a 1 000-m trip downstream?

A plane is moving due north,directly towards its destination.Its airspeed is 200 mph.A constant breeze is blowing from west to east at 20 mph.In which direction is the plane pointed?

A plane is moving due north,directly towards its destination.Its airspeed is 200 mph.A constant breeze is blowing from west to east at 40 mph.How long will it take for the plane to travel 200 miles north?

A boat travels upstream and after one hour has gone 10 km.The boat next travels downstream and after one hour has gone 14 km.If the boat's speed relative to the water is constant,what is the speed of the current in the river?

Vector 1 is 7 units long and is at 70 $\degree$ .Vector 2 is 5 units long and is at 225 $\degree$ .Vector 3 is 3 units long and is at 150 $\degree$ .Which vector has equal magnitude components?

Plane A is flying at 400 mph in the northeast direction relative to the earth.Plane B is flying at 500 mph in the north direction relative to the earth.What is the speed of Plane B as observed from Plane A?

A plane is moving due north,directly towards its destination.Its airspeed is 200 mph.A constant breeze is blowing from west to east at 30 mph.At what rate is the plane moving north?