Introduction practice
Work from memory first. Draw a diagram, state the assumptions and keep units through every step. Worked answers are separate.
· Open separate worked solutions
1. Work it out
Original textbook practice
Add A, 6 m/s at 45° counterclockwise from +x, and B, 4 m/s at 150° from +x (30° north of west). Give the result as magnitude and direction, magnitude times a unit vector, and components.
Original problem 1
Problem text
Add A, 6 m/s at 45° counterclockwise from +x, and B, 4 m/s at 150° from +x (30° north of west). Give the result as magnitude and direction, magnitude times a unit vector, and components.
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2. Work it out
Original textbook practice
Follow the diagram’s route. Its legs, starting at START, are: 2 km east, 4 km south, 5 km west, 4 km south, 5 km east, 4 km south, 5 km east, 4 km south, 10 km east, 4 km north, 5 km west, 8 km north, 5 km west, 4 km south and 2 km east. Find distance and displacement.
Original problem 2
The source answer key drops the “k” in km; use the kilometre units shown in the diagram.
Problem text
Follow the diagram’s route. Its legs, starting at START, are: 2 km east, 4 km south, 5 km west, 4 km south, 5 km east, 4 km south, 5 km east, 4 km south, 10 km east, 4 km north, 5 km west, 8 km north, 5 km west, 4 km south and 2 km east. Find distance and displacement.
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3. Work it out
Enrichment beyond the AP course
A man walks 1 km south, 3 km east and 1 km north and returns to his starting point. Give a possible starting point on an ideal spherical Earth, and explain why a flat-plane model cannot produce this result.
Original problem 3
“Where is he?” has several spherical-Earth solutions; the source key lists only the North Pole.
Problem text
A man walks 1 km south, 3 km east and 1 km north and returns to his starting point. Give a possible starting point on an ideal spherical Earth, and explain why a flat-plane model cannot produce this result.
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4. Work it out
Enrichment beyond the AP course
In a right-handed x–y plane, A has magnitude 50 m and angle +30° from +x. B has magnitude 13 m and angle −15° from +x. Find A × B.
Original problem 4
The source key gives a length unit; the cross product here has units m².
Problem text
In a right-handed x–y plane, A has magnitude 50 m and angle +30° from +x. B has magnitude 13 m and angle −15° from +x. Find A × B.
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5. Work it out
Enrichment beyond the AP course
Determine whether coordinate systems A, B, C and D in the diagram are right- or left-handed. For the perspective drawing, the diagonal axis points out toward the viewer.
Original problem 5
Problem text
Determine whether coordinate systems A, B, C and D in the diagram are right- or left-handed. For the perspective drawing, the diagonal axis points out toward the viewer.
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6. Work it out
Enrichment beyond the AP course
A 10 kg block slides down a 30° ramp. The kinetic friction coefficient is 0.4. Find the dot product of the friction force and the weight. Use g = 9.8 m/s².
Original problem 6
This is a dot product of two forces, so its unit is N²; it is not work.
Problem text
A 10 kg block slides down a 30° ramp. The kinetic friction coefficient is 0.4. Find the dot product of the friction force and the weight. Use g = 9.8 m/s².
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7. Work it out
Enrichment beyond the AP course
For real two-dimensional vectors α, β and γ, α · γ = 0 and α × β = 0. What can you conclude about β · γ? Consider separately α ≠ 0 and α = 0.
Original problem 7
The source’s unconditional answer overlooks the zero-vector case.
Problem text
For real two-dimensional vectors α, β and γ, α · γ = 0 and α × β = 0. What can you conclude about β · γ? Consider separately α ≠ 0 and α = 0.
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8. Work it out
Original textbook practice
Use the pictured path: 2 km east, 8 km south, 5 km west, 4 km south, 10 km west, 8 km north, 5 km east, 4 km south and 3 km east. Define route efficiency as η = |displacement| / distance traveled. Find η.
Original problem 8
Use the magnitude of displacement in this scalar ratio; a vector divided by distance would remain a vector.
Problem text
Use the pictured path: 2 km east, 8 km south, 5 km west, 4 km south, 10 km west, 8 km north, 5 km east, 4 km south and 3 km east. Define route efficiency as η = |displacement| / distance traveled. Find η.
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9. Work it out
Enrichment beyond the AP course
The diagram is a closed track with four semicircular turns. Correct radius A to 30 m; retain B = C = D = 10 m. The two outer straight segments are each 250 m and the two inner straight segments are each 150 m. Find displacement and distance for one complete lap.
Original problem 9
The original A = 75 m cannot connect the depicted three 10 m semicircles with horizontal straights: vertical closure requires 2RA = 2(RB + RC + RD). This reviewed version repairs A to 30 m. Taking the original arc radii independently would give 1129.87 m, but that geometry would not close.
Problem text
The diagram is a closed track with four semicircular turns. Correct radius A to 30 m; retain B = C = D = 10 m. The two outer straight segments are each 250 m and the two inner straight segments are each 150 m. Find displacement and distance for one complete lap.
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10. Work it out
Enrichment beyond the AP course
Embed the following vectors in the x–y plane of a right-handed 3D system: α = (4, 3), β = (1, 3), γ = (−1, 4), δ = (8, 0). Evaluate (α × β) · (γ × δ).
Original problem 10
Direct component calculation corrects the source key’s −387.6.
Problem text
Embed the following vectors in the x–y plane of a right-handed 3D system: α = (4, 3), β = (1, 3), γ = (−1, 4), δ = (8, 0). Evaluate (α × β) · (γ × δ).
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Additional practice
11. What needs a direction?
Original textbook practice
Classify mass, temperature, displacement, velocity, kinetic energy and force as scalars or vectors. Explain why a negative temperature is still a scalar.
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12. A force pointing west
Original textbook practice
With east as +x, a horizontal force is Fₓ = −12 N. State its magnitude, direction and unit vector.
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13. Choose the quadrant
Original textbook practice
A displacement is (−6, 8) m. Find its magnitude and its direction counterclockwise from +x.
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14. From an angle to components
Original textbook practice
A velocity has magnitude 12 m/s and direction 120° counterclockwise from +x. Find vₓ and vᵧ.
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15. The zero vector
Original textbook practice
Two displacement vectors are (3, −4) m and (−3, 4) m. Find their sum, magnitude and unit vector.
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16. A direction in three dimensions
Enrichment beyond the AP course
Normalize the vector (2, −3, 6). Verify that your result has magnitude 1.
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17. An orthogonality test
Enrichment beyond the AP course
Are A = (3, 4) and B = (8, −6) perpendicular? Support your answer with a dot product.
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18. Angle between two vectors
Enrichment beyond the AP course
Find the smaller angle between A = (1, 2) and B = (2, 1).
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19. Project a pull
Original textbook practice
A 50 N rope force acts at 37° above a horizontal floor. Find its horizontal and vertical components.
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20. Cross product and handedness
Enrichment beyond the AP course
In a right-handed coordinate system, A = (2, 0, 0) m and B = (0, 3, 0) m. Find A × B and B × A.
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21. A parallelogram from vectors
Enrichment beyond the AP course
A = (4, 1, 0) m and B = (−2, 3, 0) m form adjacent sides of a parallelogram. Find its area.
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22. Work is a scalar
Original textbook practice
A constant force F = (6, −2) N acts through displacement Δr = (3, 4) m. Find the work done.
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23. Two routes, one endpoint
Original textbook practice
A walker goes 300 m east and then 400 m north. Another walker goes directly between the same endpoints. Compare the distances and displacements.
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24. A return journey
Original textbook practice
A runner travels 100 m east in 20 s and returns in 30 s. Find average speed and average velocity for the whole trip.
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25. Boat and current
Enrichment beyond the AP course
A boat moves north at 3 m/s relative to the water. The water flows east at 4 m/s. Find the boat’s velocity relative to the bank.
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26. Net force in a plane
Original textbook practice
A 2 kg cart experiences forces (8, 0) N, (−2, 6) N and (0, −2) N. Find its acceleration.
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27. Action and reaction
Original textbook practice
A person pushes a wall with 80 N. Identify the Newton’s third-law pair and explain why the two forces do not cancel in the person’s free-body diagram.
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28. An accelerating elevator
Original textbook practice
A 60 kg passenger accelerates upward at 1.5 m/s². Find the normal force from the floor. Use g = 9.8 m/s².
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29. Balanced ropes
Original textbook practice
A 100 N lamp hangs symmetrically from two ropes, each at 30° above horizontal. Find the tension in each rope.
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30. Can magnitudes alone determine a sum?
Original textbook practice
Two forces have magnitudes 5 N and 7 N, but their directions are unknown. What are the largest and smallest possible magnitudes of their sum? Give configurations that achieve each.
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