Question 1

A plane is moving due north,directly towards its destination.Its airspeed is 230 mph.A constant breeze is blowing from west to east at 30 mph.In which direction is the plane pointed?
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A) 7.4° W of N
B) 82.6° W of N
C) 44.8° W of N
D) 82.5° E of N
E) ​45.2° W of N

Accepted Answer

The plane must compensate for the eastward wind by pointing slightly west of north. Using vector addition and trigonometry, the angle can be calculated as arctan(30/230) ≈ 7.4° west of north.

Question 2

A boat travels upstream and after one hour has gone 8 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?
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A) 2 km/h
B) 3 km/h
C) 4 km/h
D) 5 km/h
E) ​6 km/h

Accepted Answer

The boat's speed relative to the water is constant. Let the speed of the boat be 'b' km/h and the speed of the current be 'c' km/h. Upstream, the effective speed is (b - c) km/h, and downstream, it is (b + c) km/h. Given that upstream the boat travels 8 km in 1 hour, we have b - c = 8. Downstream, the boat travels 14 km in 1 hour, so b + c = 14. Solving these two equations simultaneously:  b - c = 8  b + c = 14  Adding the equations: 2b = 22, so b = 11.  Substituting b = 11 into b - c = 8 gives 11 - c = 8, so c = 3 km/h.

Question 3

A boat moves at 12 m/s relative to the water.If the boat is in a river where the current is 2 m/s,how long does it take the boat to make a complete round trip of 1,100 m upstream followed by a 1,100-m trip downstream?
A) 220 s
B) 157.1 s
C) 188.6 s
D) 267.1 s
E) ​298.6 s

Accepted Answer

To find the time taken, we need to first find the speed of the boat with respect to the ground. 
Upstream speed = boat speed - current speed = 12 - 2 = 10 m/s 
Downstream speed = boat speed + current speed = 12 + 2 = 14 m/s

Time taken to travel upstream = distance/speed = 1100/10 = 110 seconds 
Time taken to travel downstream = distance/speed = 1100/14 = 78.57 seconds (approx.)

Total time taken = time upstream + time downstream = 110 + 78.57 = 188.57 seconds (approx.)

Therefore, the answer is C.

Question 4

Plane A is flying at 440 mph in the northeast direction relative to the earth.Plane B is flying at 550 mph in the north direction relative to the earth.What is the speed of Plane B as observed from Plane A?
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A) 570 mph
B) 916 mph
C) 392 mph
D) 704 mph
E) ​714 mph

Accepted Answer

First, we must find the vectors of each plane's velocity. Let us use the origin as the starting point of the two planes, for simplicity.

Plane A is traveling at 440 mph in the northeast direction. This can be represented by a vector with a magnitude of 440 mph and an angle of 45 degrees (northeast is halfway between north and east).

Plane B is traveling at 550 mph in the north direction. This can be represented by a vector with a magnitude of 550 mph and an angle of 90 degrees.

To find the relative velocity of Plane B as observed from Plane A, we simply need to subtract the vectors pointing from the origin to each plane.

Using trigonometry, we can find that the components of Plane A's velocity vector are 311.1 mph to the east and 311.1 mph to the north (this can be found by multiplying the magnitude of 440 by cos(45) and sin(45), respectively).

Similarly, the components of Plane B's velocity vector are 0 mph to the east and 550 mph to the north.

Subtracting the two vectors, we get a result of 311.1 mph to the east and 238.9 mph to the north.

Using the Pythagorean theorem, we find that the magnitude of this vector is approximately 392 mph.

Therefore, the speed of Plane B as observed from Plane A is approximately 392 mph, which corresponds to choice C.

Question 5

A plane is moving due north,directly towards its destination.Its airspeed is 240 mph.A constant breeze is blowing from west to east at 20 mph.How long will it take for the plane to travel 240 miles north?
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A) one hour
B) more than one hour
C) less than one hour
D) more information is needed
E) ​none of the above

Accepted Answer

The answer of A plane is moving due north,directly towards...

Question 6

A plane is moving due north,directly towards its destination.Its airspeed is 240 mph.A constant breeze is blowing from west to east at 50 mph.At what rate is the plane moving north?
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A) 235 mph
B) 245 mph
C) 230 mph
D) 225 mph
E) ​220 mph

Accepted Answer

The plane's airspeed is 240 mph due north, and the wind is blowing from west to east at 50 mph. This means that the plane is actually moving at a speed of 240 mph in the direction of 240 degrees (due north). The wind is blowing at a speed of 50 mph in the direction of 270 degrees (due east). To find the rate at which the plane is moving north, we need to find the component of the plane's velocity in the north direction. This can be done using the formula:```V_north = V * cos(theta)```where:* V is the speed of the plane* theta is the angle between the plane's velocity and the north directionIn this case, V = 240 mph and theta = 240 degrees. Plugging these values into the formula, we get:```V_north = 240 mph * cos(240 degrees) = 235 mph```Therefore, the plane is moving north at a rate of 235 mph.

Question 7

Plane A is flying at 350 mph in the northeast direction relative to the earth.Plane B is flying at 520 mph in the north direction relative to the earth.What is the direction of motion of Plane B as observed from Plane A?
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A) 72.1° N of E
B) 42.2° N or W
C) 47.8° N of W
D) 17.9° N of W
E) ​41.1° N of W

Accepted Answer

We need to find the relative motion of Plane B as observed from Plane A. To do this, we need to find the velocity vector of Plane B from the perspective of Plane A.

Let's assume that the velocity of Plane B is made up of two components - one towards the north (as given in the problem) and the other towards the east.

The velocity of Plane B towards east (relative to the earth) is zero because it is not moving eastward. The velocity of Plane A towards east (relative to the earth) is 350 mph towards northeast, which means that the velocity towards the east is 350/sqrt(2) mph.

Therefore, the velocity vector of Plane B as observed from Plane A is the vector sum of its velocity towards north (520 mph) and its velocity towards east (350/sqrt(2) mph).

Using vector addition, we get:

Velocity vector of Plane B as observed from Plane A = sqrt[(520)^2 + (350/sqrt(2))^2] = 605.1 mph

Now, we need to find the angle between this velocity vector and the north-east direction (the direction of Plane A's motion).

Using trigonometry, we get:

tan(theta) = (350/sqrt(2)) / 520 
=> theta = 47.8°

Therefore, the direction of motion of Plane B as observed from Plane A is 47.8° N of W.

Question 8

Vector 1 is 8 units long and is at 225°.Vector 2 is 5 units long and is at 120°.Vector 3 is 4 units long and is at 70°.Which vector has equal magnitude components?
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A) Vector 3
B) Vector 1
C) Vector 2
D) None of the vectors has equal magnitude components.
E) ​Vector 2 and Vector 3

Accepted Answer

The answer of Vector 1 is 8 units long and...

Question 9

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

Accepted Answer

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

Question 10

A river flows due east at 3 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 15 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?
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A) 120 m
B) 50 m
C) 45 m
D) 60 m
E) ​70 m

Accepted Answer

We can use the formula:

distance = rate × time

First, we need to find the time it takes for the boat to cross the river. We can use the fact that the distance across the river is 300 meters and the boat's velocity relative to the water is 15 m/s:

time = distance / velocity
time = 300 m / 15 m/s
time = 20 s

Now we can use the boat's velocity and the time it took to cross the river to find how far downstream the boat moved:

distance downstream = velocity downstream × time
velocity downstream = 3 m/s (since the river flows due east)
time = 20 s
distance downstream = 3 m/s × 20 s
distance downstream = 60 m

Therefore, the boat moved 60 meters downstream by the time it reached the far shore. The correct answer is D) 60 m.

Question 11

V,,Have magnitudes 8,13,and 22.When these vectors are added,what is the least possible magnitude of their resultant?
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A) 43
B) 0
C) 14
D) 1
E) ​23

Accepted Answer

The answer of V,,Have magnitudes 8,13,and 22.When these vectors are...

Question 12

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 13

Four vectors all have the same magnitude.Vector 1 is at 60°,Vector 2 is at 135°,vector 3 is at 210°,and Vector 4 is at 315°.Which vector has the greatest magnitude x-component and which vector has the greatest magnitude y-component?
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A) Vector 1,Vector 2
B) Vector 3,Vector 4
C) Vector 1,Vector 3
D) Vector 3,Vector 2
E) ​Vector 2,Vector 4

Accepted Answer

To find which vector has the greatest magnitude x-component, we need to look at the cosine values of each angle (cosine is the x-component of a vector with that angle). The cosine values are as follows:
Vector 1: cos(60°) = 0.5
Vector 2: cos(135°) = -0.707
Vector 3: cos(210°) = -0.5
Vector 4: cos(315°) = 0.707
Therefore, vectors 1 and 4 have the greatest magnitude x-components. 
To find which vector has the greatest magnitude y-component, we need to look at the sine values of each angle (sine is the y-component of a vector with that angle). The sine values are as follows:
Vector 1: sin(60°) = 0.866
Vector 2: sin(135°) = 0.707
Vector 3: sin(210°) = -0.866
Vector 4: sin(315°) = -0.707
Therefore, vectors 1 and 3 have the greatest magnitude y-components.
So the answer is C, Vector 1 and Vector 3.

Question 14

A projectile is fired at an angle of elevation of 45°.Neglecting air resistance,what are possible angles in flight between the acceleration vector and the velocity vector?
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A) 160° and 40°
B) 20° and 70°
C) 90° and 45°
D) none of the above
E) ​10° and 45°

Accepted Answer

When a projectile is fired at an angle of 45 degrees, the velocity vector is split equally into horizontal and vertical components. As air resistance is neglected, the acceleration vector is equal to the gravitational force acting downwards. During the projectile's motion, the direction of the velocity vector changes continuously, while the acceleration vector remains constant. At the highest point of the projectile's path, the velocity vector is perpendicular to the acceleration vector. Thus, the only possible angles between the acceleration and velocity vectors are 90 degrees and 45 degrees. Therefore, the correct choice is (C) 90° and 45°.

Quiz 3: Vectors and Two-Dimensional Motion

A plane is moving due north,directly towards its destination.Its airspeed is 230 mph.A constant breeze is blowing from west to east at 30 mph.In which direction is the plane pointed?

A boat travels upstream and after one hour has gone 8 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 boat moves at 12 m/s relative to the water.If the boat is in a river where the current is 2 m/s,how long does it take the boat to make a complete round trip of 1,100 m upstream followed by a 1,100-m trip downstream?

Plane A is flying at 440 mph in the northeast direction relative to the earth.Plane B is flying at 550 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 240 mph.A constant breeze is blowing from west to east at 20 mph.How long will it take for the plane to travel 240 miles north?

A plane is moving due north,directly towards its destination.Its airspeed is 240 mph.A constant breeze is blowing from west to east at 50 mph.At what rate is the plane moving north?

Plane A is flying at 350 mph in the northeast direction relative to the earth.Plane B is flying at 520 mph in the north direction relative to the earth.What is the direction of motion of Plane B as observed from Plane A?

Vector 1 is 8 units long and is at 225°.Vector 2 is 5 units long and is at 120°.Vector 3 is 4 units long and is at 70°.Which vector has equal magnitude components?

A river flows due east at 3 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 15 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,,andHave magnitudes 8,13,and 22.When these vectors are added,what is the least possible magnitude of their resultant?

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?

Four vectors all have the same magnitude.Vector 1 is at 60°,Vector 2 is at 135°,vector 3 is at 210°,and Vector 4 is at 315°.Which vector has the greatest magnitude x-component and which vector has the greatest magnitude y-component?

A projectile is fired at an angle of elevation of 45°.Neglecting air resistance,what are possible angles in flight between the acceleration vector and the velocity vector?

Quiz 3: Vectors and Two-Dimensional Motion

A plane is moving due north,directly towards its destination.Its airspeed is 230 mph.A constant breeze is blowing from west to east at 30 mph.In which direction is the plane pointed? ​

A boat travels upstream and after one hour has gone 8 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 boat moves at 12 m/s relative to the water.If the boat is in a river where the current is 2 m/s,how long does it take the boat to make a complete round trip of 1,100 m upstream followed by a 1,100-m trip downstream?

Plane A is flying at 440 mph in the northeast direction relative to the earth.Plane B is flying at 550 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 240 mph.A constant breeze is blowing from west to east at 20 mph.How long will it take for the plane to travel 240 miles north? ​

A plane is moving due north,directly towards its destination.Its airspeed is 240 mph.A constant breeze is blowing from west to east at 50 mph.At what rate is the plane moving north? ​

Plane A is flying at 350 mph in the northeast direction relative to the earth.Plane B is flying at 520 mph in the north direction relative to the earth.What is the direction of motion of Plane B as observed from Plane A? ​

Vector 1 is 8 units long and is at 225°.Vector 2 is 5 units long and is at 120°.Vector 3 is 4 units long and is at 70°.Which vector has equal magnitude components? ​

A river flows due east at 3 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 15 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,,andHave magnitudes 8,13,and 22.When these vectors are added,what is the least possible magnitude of their resultant? ​