Question 1

Use the component method of vector addition to find the components of the resultant of the four displacements shown in the figure. The magnitudes of the displacements are: A = 2.25 cm, B = 6.35 cm, C = 5.47 cm, and D = 4.19 cm.  x component y component

Accepted Answer

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Question 2

A displacement vector is 23 km in length and directed 65° south of east. What are the components of this vector? Eastward Component Southward Component

Accepted Answer

The eastward component can be found using cosine function. 
Eastward component = 23 km × cos 65° = 9.7 km
The southward component can be found using sine function. 
Southward component = 23 km × sin 65° = 21 km
Therefore, the answer is D) 9.7 km and 21 km.

Question 3

A sailboat leaves a harbor and sails 1.8 km in the direction 65° south of east, where the captain stops for lunch. A short time later, the boat sails 1.1 km in the direction 15° north of east. What is the magnitude of the resultant displacement from the harbor?

Accepted Answer

To find the resultant displacement, we need to break each leg of the journey into its east and north components, sum these components, and then use the Pythagorean theorem. The resultant displacement is approximately 2.3 km.

Question 4

During the execution of a play, a football player carries the ball for a distance of 33 m in the direction 58° north of east. To determine the number of meters gained on the play, find the northward component of the ball's displacement.

Accepted Answer

The answer of During the execution of a play, a...

Question 5

Three vectors , , and have the following x and y components: A_x = 1 m, A_y = 0 m, B_x = 1 m, B_y = 1 m, C_x = 0 m, C_y = -1 m According to the graph, how are , , and combined to result in the vector ?

Accepted Answer

To get the vector 11ea7e4c_68c9_5320_9a0a_d77403f22ca5_TB3413_11, we need to add the three given vectors. Looking at the given choices, we see that only Choice C shows the addition of three vectors, which results in the required vector. Therefore, the best choice is C.

Question 6

A race car will make one lap around a circular track of radius R. When the car has traveled halfway around the track, what is the magnitude of the car's displacement from the starting point?

Accepted Answer

When the car has traveled halfway around the circular track, it is at the point directly opposite the starting point on the circle. The displacement is the straight-line distance between the starting point and this point, which is the diameter of the circle, equal to 2R.

Question 7

A displacement vector has a magnitude of 810 m and points at an angle of 18° above the positive x axis. What are the x and y scalar components of this vector? x scalar component y scalar component

Accepted Answer

The x component is the magnitude times the cosine of the angle, and the y component is the magnitude times the sine of the angle. So, the x component is 810 m * cos(18°) = 770 m, and the y component is 810 m * sin(18°) = 250 m.

Question 8

A bird flies 25.0 m in the direction 55° east of south to its nest on a cliff. The bird then flies 75.0 m in the direction 55° west of north to the top of a tree. What are the northward and westward components of the resultant displacement of the bird from its nest? northward westward

Accepted Answer

The northward component is the sum of the northward components of the two displacements:$$25.0 \sin 55^\circ + 75.0 \sin 55^\circ = 29.0 	ext{ m}$$The westward component is the sum of the westward components of the two displacements:$$25.0 \cos 55^\circ + 75.0 \cos 55^\circ = 41.0 	ext{ m}$$

Question 9

A vector  has a magnitude of 40.0 units and points 35.0° above the positive x axis. A second vector  has a magnitude of 65.0 units and points in the negative x direction. Use the component method of vector addition to find the magnitude and direction, relative to the positive x axis, of the resultant  =  +  .

Accepted Answer

The answer of A vector  has a magnitude of...

Question 10

City A lies 30 km directly south of city B. A bus, beginning at city A travels 50 km at 37° north of east to reach city C. How far and in what direction must the bus go from city C to reach city B?

Accepted Answer

Using vector addition, we can draw a diagram to represent the distances and directions traveled by the bus:
     C
     /   
 50/  37°\
 /        \
A---------B
We can see that the bus has traveled 30 km south and 50 km at a 37° angle north of east, which can be broken down into horizontal and vertical components:
Horizontal: 50 cos(37°) = 39.6 km east
Vertical: 50 sin(37°) = 30 km north
Therefore, city C is 39.6 km east and 30 km north of city A. To get to city B, the bus must travel directly west, which is the opposite direction of the vector from C to A. Using the Pythagorean theorem, we can find the distance from C to B:
distance = sqrt((39.6 km)^2 + (30 km)^2) = 49.8 km
Therefore, the bus must go 49.8 km west from city C to reach city B, which is closest to choice B (40 km, west).

Question 11

Use the component method of vector addition to find the resultant of the following three vectors:  = 56 km, east  = 11 km, 22° south of east  = 88 km, 44° west of south

Accepted Answer

The answer of Use the component method of vector addition...

Question 12

Two vectors and , are added together to form the vector = + . The relationship between the magnitudes of these vectors is given by: C_x= A cos 30° + B and C_y= -A sin 30°. Which statement best describes the orientation of these vectors?

Accepted Answer

We know that the magnitudes of the vectors can be represented as: C_x= A cos 30° + B C_y= -A sin 30° Since the vector 11ea7e4c_68ca_b2d1_9a0a_ab3081cf4f31_TB3413_11 is the sum of 11ea7e4c_68ca_b2d2_9a0a_a552a75d3bec_TB3413_11 and 11ea7e4c_68ca_b2d3_9a0a_5fe64bfe9b3e_TB3413_11, the angle between the two vectors can be found using the dot product formula: 11ea7e4c_68ca_b2d2_9a0a_a552a75d3bec_TB3413_11 * 11ea7e4c_68ca_b2d3_9a0a_5fe64bfe9b3e_TB3413_11 = |11ea7e4c_68ca_b2d2_9a0a_a552a75d3bec_TB3413_11| * |11ea7e4c_68ca_b2d3_9a0a_5fe64bfe9b3e_TB3413_11| * cos θ θ = 30° Since the x-component of 11ea7e4c_68ca_b2d1_9a0a_ab3081cf4f31_TB3413_11 is positive and the y-component is negative, it points 30° below the positive x axis. Therefore, the orientation of 11ea7e4c_68ca_b2d2_9a0a_a552a75d3bec_TB3413_11 is also 30° below the positive x axis. The other vector, 11ea7e4c_68ca_b2d3_9a0a_5fe64bfe9b3e_TB3413_11, points in the positive x direction. Thus, the correct choice is C.

Question 13

Town A lies 20 km north of town B. Town C lies 13 km west of town A. A small plane flies directly from town B to town C. What is the displacement of the plane?

Accepted Answer

The answer of Town A lies 20 km north of...

Question 14

Which one of the following statements concerning vectors and scalars is false?

Accepted Answer

A vector that is zero must have components that are also zero.

Question 15

A runaway dog walks 0.64 km due north. He then runs due west to a hot dog stand. If the magnitude of the dog's total displacement vector is 0.91 km, what is the magnitude of the dog's displacement vector in the due west direction?

Accepted Answer

The answer of A runaway dog walks 0.64 km due...

Question 16

The x and y components of a displacement vector are -3.00 m and +4.00 m, respectively. What angle does this vector make with the positive x axis?

Accepted Answer

The answer of The x and y components of a...

Question 17

An escaped convict runs 1.70 km due east of the prison. He then runs due north to a friend's house. If the magnitude of the convict's total displacement vector is 2.50 km, what is the direction of his total displacement vector with respect to due east?

Accepted Answer

The answer of An escaped convict runs 1.70 km due...

Question 18

A force,  , of magnitude 2.0 N and directed due east is exerted on an object. A second force exerted on the object is  = 2.0 N, due north. What is the magnitude and direction of a third force,  , which must be exerted on the object so that the resultant force is zero?

Accepted Answer

The answer of A force,  , of magnitude 2.0...

Question 19

A bug crawls 2.25 m along the base of a wall. Upon reaching a corner, the bug's direction of travel changes from south to west. The bug then crawls 3.15 m before stopping. What is the magnitude of the bug's displacement?

Accepted Answer

The bug's displacement can be found using the Pythagorean theorem. The displacement is the hypotenuse of a right triangle with legs of 2.25 m and 3.15 m. Calculating: √(2.25² + 3.15²) = √(5.0625 + 9.9225) = √14.985 = 3.87 m.

Question 20

Four members of the Main Street Bicycle Club meet at a certain intersection on Main Street. The members then start from the same location, but travel in different directions. A short time later, displacement vectors for the four members are:  = 2.0 km, east;  = 5.2 km, north;  = 4.9 km, west;  = 3.0 km, south What is the resultant displacement  of the members of the bicycle club:  =  +  +  +  ?

Accepted Answer

The answer of Four members of the Main Street Bicycle...

Quiz 1: Introduction and Mathematical Concepts

Use the component method of vector addition to find the components of the resultant of the four displacements shown in the figure. The magnitudes of the displacements are: A = 2.25 cm, B = 6.35 cm, C = 5.47 cm, and D = 4.19 cm. x component y component