4 questions, one for each idea where we can. Answer them, then see which ideas to fix.
Question 1 of 4
Two carts move along the same straight track, with the positive direction to the right. Cart 1 has velocity +3 m/s and cart 2 has velocity −3 m/s. Which statement correctly compares the two carts?
Answer and reasoning
ATheir velocities are equal, since each of the two velocities has the same magnitude, 3 m/s. A student who compares vectors by magnitude alone picks this. Two vectors are equal only if they have the same magnitude AND the same direction; these velocities point in opposite directions.
BThe speeds of carts 1 and 2 are equal, but their velocities differ in direction.Correct Speed is a scalar, the magnitude of velocity, so both carts move at 3 m/s. Velocity is a vector: the opposite signs show that the carts move in opposite directions, so their velocities are different.
CCart 2 has the smaller speed, since its velocity is a value less than zero. A student who reads −3 m/s as a smaller quantity than +3 m/s picks this. The minus sign gives only the direction; the magnitude of −3 m/s is 3 m/s, so the two speeds are equal.
DCart 2 is slowing down but cart 1 is not, since cart 2's velocity is the negative one. A student who reads a negative velocity as slowing down picks this. The sign says only that cart 2 moves to the left; one velocity value for each cart says nothing about whether either speed is changing.
Working Speeds: |+3 m/s| = |−3 m/s| = 3 m/s, so the speeds are equal. Velocities: +3 m/s (to the right) and −3 m/s (to the left) have the same magnitude but opposite directions, so the velocities differ.
In the diagram, the top arrow represents a velocity of +2 m/s, with the positive direction to the right. Arrows A–D are drawn to the same scale. Which arrow represents a velocity of −6 m/s?
Answer and reasoning
AArrow A A student who drops the minus sign picks this. This arrow has the right length, three times the reference, but it points to the right, so it represents +6 m/s. In one dimension the sign of a component is its direction.
BArrow B A student who thinks an arrow's length is arbitrary picks this. It points the right way but is the same length as the +2 m/s arrow, so it represents −2 m/s; a 6 m/s velocity needs an arrow three times as long.
CArrow CCorrect −6 m/s has a magnitude of 6 m/s, three times the magnitude of the +2 m/s reference, and its minus sign means it points in the negative direction. The correct arrow points to the left and is three times as long as the reference arrow.
DArrow D A student who thinks −6 m/s is a smaller quantity than +2 m/s picks this. It points to the left but is shorter than the reference, so it represents only −1 m/s. The magnitude of −6 m/s is 6 m/s, three times the reference.
Working |−6 m/s| ÷ |+2 m/s| = 3, so the arrow must be three times as long as the reference arrow; the minus sign means it points in the negative direction, to the left. Only C is three times as long and points left.
A student records four facts about a school bus trip. Which fact describes a vector quantity?
Answer and reasoning
AThe whole trip lasted for a total time of 2 hours. A student who thinks time is a vector because it 'runs forward' picks this. Two hours is described completely by a magnitude and a unit; it has no direction in space, so time is a scalar.
BThe temperature outside at noon was −3 °C. A student who thinks any quantity that can be negative is a vector picks this. −3 °C is a value below the zero of a temperature scale, not a direction in space; temperature is a scalar.
CThe bus's speedometer read 60 km/h at noon. A student who treats speed and velocity as the same quantity picks this. A speedometer shows speed, the magnitude of the velocity with no direction, so its reading is a scalar.
DThe bus ended the trip 30 km east of the school.Correct '30 km east of the school' gives a magnitude (30 km) and a direction (east) measured from a chosen origin, the school. It is the bus's final position, a vector.
Working 30 km east of the school has a magnitude and a direction relative to an origin (a position). 2 hours (time), −3 °C (temperature) and 60 km/h (a speed) have magnitudes only.
A robot moves along a straight track. It makes displacement A and then displacement B, shown in the diagram drawn tip to tail to scale; the spacing of the grid lines represents 10 m. Taking the positive direction to the right, what is the robot's resultant displacement?
Answer and reasoning
A+80 m A student who adds the magnitudes, 30 m + 50 m, picks this. That is the distance the robot travels. A and B point in opposite directions, so their components have opposite signs and partly cancel.
B+20 m A student who subtracts the smaller magnitude from the larger and drops the sign picks this. The larger displacement, B, points to the left, so the resultant points to the left: −20 m, not +20 m.
C−20 mCorrect Give each displacement the sign of its direction and add: A = +30 m and B = −50 m, so the resultant is +30 m + (−50 m) = −20 m. The robot ends 20 m to the left of its starting point.
D−80 m A student who subtracts instead of adding, B − A = −50 m − 30 m, picks this. That is the difference between the two displacements; the resultant is their sum, A + B.
Working A = +30 m (3 grid spacings to the right); B = −50 m (5 grid spacings to the left). Resultant = (+30 m) + (−50 m) = −20 m, that is, 20 m to the left of the start.
In preparation: 0 of 4 sections compiled and reviewed. The rest show key terms and common mistakes from our question bank until they are.
1.1.A.1 Scalar quantity Fix
Scalar quantity
A quantity described completely by a magnitude (a number with a unit) and no direction. Examples: distance (m), speed (m/s), time (s), mass (kg) and temperature. A scalar measured from a chosen zero, such as a temperature, can be negative without having a direction.
Vector quantity
A quantity described by both a magnitude and a direction, such as 30 m east or 4 m/s in the −x direction. Two vectors are equal only if both their magnitudes and their directions are the same.
Magnitude
The size of a quantity, with its unit and without any direction. The magnitude of a vector is never negative: the magnitude of a velocity of −4 m/s is 4 m/s, the speed.
Students often think Time is a vector, because it moves in one direction (forward) and is described with words such as 'before' and 'after'. In fact No. Time is a scalar: 'two hours' is described completely by a magnitude and a unit. 'Forward in time' is not a direction in space that an arrow could represent.
Students often think Any quantity that can take a negative value has a direction and is a vector, so a temperature of −3 °C is a vector, and a signed component such as vx must be written with a vector arrow. In fact No. A scalar measured from a chosen zero can be negative: a temperature of −3 °C is a value below the zero of a scale, not a direction. A component such as vx is signed, but it is a number describing a vector along an axis, and AP notation gives it no arrow.
1.1.A.2 Vector arrow (vector diagram) Fix
Vector arrow (vector diagram)
A vector is modeled as an arrow that points in the vector's direction and whose length is proportional to its magnitude, using one scale for every arrow in the diagram: an arrow twice as long represents a vector of twice the magnitude.
Students often think A vector arrow shows only a direction and its length is arbitrary, so arrows of different lengths can represent equal magnitudes, and any arrow pointing the right way represents the vector. In fact Yes. In a vector diagram an arrow points in the vector's direction and its length is proportional to the vector's magnitude, so an arrow twice as long represents a vector of twice the magnitude.
1.1.A.3 Distance traveled Fix
Distance traveled
A scalar: the total length of the path an object follows, whatever its direction of travel. SI unit: m.
Speed
A scalar: how fast an object is moving, with no direction; the magnitude of its velocity. SI unit: m/s.
Position
A vector: an object's location relative to a chosen origin, given in one dimension as a signed coordinate such as x = −15 m. SI unit: m.
Displacement
A vector: the change in an object's position, from where it starts to where it ends. It depends only on those two positions, not on the path. SI unit: m.
Velocity
A vector: how fast and in which direction an object's position is changing. Its magnitude is the speed. SI unit: m/s.
Acceleration
A vector: how quickly and in which direction an object's velocity is changing. Its direction need not be the direction of motion. SI unit: m/s².
Vector notation
A vector is written with an arrow above its symbol, such as v⃗ or a⃗, as in v⃗ = v⃗0 + a⃗t. The same letter without an arrow (v) is used for the vector's magnitude.
Component along an axis
The part of a vector along a chosen axis, written with a subscript and no arrow (vx, ax, Δx). In one dimension it is a signed number: the sign completely describes the direction and the number without its sign is the magnitude.
Students often think A negative vector component is a smaller quantity than a positive one, as on a number line: −5 m/s is less motion than +2 m/s, and a vector in the negative direction is drawn with a shorter arrow. In fact No. In one dimension the sign of a component gives only its direction. A velocity of −5 m/s is a greater speed than +2 m/s, and an arrow for −6 m/s is three times as long as an arrow for +2 m/s.
Students often think A minus sign on a velocity or an acceleration means that the object is slowing down, so a negative velocity or a negative acceleration is a 'deceleration'. In fact No. A negative component means only that the vector points in the negative direction of the chosen axis. An object with a negative velocity is moving in the −x direction, and an object with a negative acceleration speeds up if it is also moving in the −x direction.
1.1.B.1 One-dimensional coordinate system Fix
One-dimensional coordinate system
A straight axis with a chosen origin and a chosen positive direction. The choice of positive direction is free, but once made it applies to every vector in the problem; the opposite direction is negative.
Vector sum (resultant) in one dimension
The single vector equivalent to two or more vectors combined. In one dimension it is found by adding the signed components: +30 m and −50 m give a resultant of −20 m. Drawn tip to tail, it runs from the tail of the first arrow to the tip of the last.
Students often think Vectors are combined by adding their magnitudes, whatever their directions, so +30 m and −50 m make a total of 80 m. In fact No. In a one-dimensional coordinate system vectors in opposite directions have components of opposite sign, and the vector sum is the sum of the signed components: +30 m and −50 m give −20 m, not 80 m.
Students often think Only the size of a vector matters: opposite vectors are combined by subtracting the smaller magnitude from the larger and giving the answer as positive, and a minus sign can be dropped, whatever the direction. In fact No. In one dimension the sign is the direction, and dropping it changes the vector. A resultant of −20 m is 20 m in the negative direction; reporting it as +20 m describes a different displacement.
5 more questions. Every wrong answer here is a real mistake students make, and you see why it is wrong as soon as you answer.
Question 1 of 5
The diagram shows arrows representing the velocities of two cars, X and Y, on a straight road at the same instant, drawn to the same scale. The spacing of the grid lines represents 1 m/s. Which claim is supported by the diagram?
Answer and reasoning
AX is moving twice as fast as Y, in the opposite direction to Y.Correct Arrow lengths are proportional to magnitudes. X's arrow spans 4 grid spacings (4 m/s) to the left and Y's spans 2 grid spacings (2 m/s) to the right, so X's speed is twice Y's and the two velocities point in opposite directions.
BX and Y are moving equally fast; the arrows differ only in direction. A student who thinks an arrow's length carries no meaning picks this. In a vector diagram length is proportional to magnitude: X's arrow is twice as long as Y's, so X moves twice as fast.
CY is moving faster than X, since X points in the negative direction. A student who treats a vector in the negative direction as a smaller quantity picks this. Direction does not affect size: X's arrow is the longer one, so X is the faster car.
DX is slowing down, since its velocity points in the negative (−x) direction. A student who reads a velocity in the −x direction as slowing down picks this. A leftward arrow means only that X moves to the left; the diagram shows each car's velocity at one instant and says nothing about whether either speed is changing.
Working X spans 4 grid spacings to the left: 4 m/s in the −x direction (vx = −4 m/s). Y spans 2 grid spacings to the right: 2 m/s in the +x direction (vx = +2 m/s). X's speed is twice Y's, and the directions are opposite.
A cart moves along a straight track that lies along the x axis. A textbook writes the cart's velocity as v⃗ in one equation and as vx in another. Which statement about these two symbols is correct?
Answer and reasoning
Av⃗ is the velocity vector; vx is its component along x, whose sign gives the direction.Correct The arrow marks v⃗ as a vector, with a magnitude and a direction. For motion along one axis the vector is described by its component vx, a signed number whose sign gives the direction, so vx needs no arrow.
Bv⃗ and vx name the same quantity, so the arrow may be written or left off at will. A student who thinks the arrow is decoration picks this. The arrow marks a vector; vx is a component, a signed number along one axis. Writing v⃗ = 12 m/s, for example, leaves out the direction.
Cvx is the magnitude of v⃗, so it is the cart's speed and cannot have a negative value. A student who treats a component as a magnitude picks this. The magnitude of v⃗, the speed, is never negative, but the component vx is negative whenever the cart moves in the −x direction.
Dvx can take negative values, so it is a vector and has to be written with an arrow. A student who thinks every signed quantity needs vector notation picks this. vx does take negative values, but it is a component along an axis, and vector notation is not required for components: the sign alone gives the direction.
A cart moves along a straight track, with the positive direction to the right. At a certain instant the cart's velocity is vx = −4 m/s. Which statement describes the cart's motion at that instant?
Answer and reasoning
AIt is moving to the right but is slowing down. A student who reads a negative velocity as slowing down picks this. A negative velocity means motion in the negative direction, to the left, and a single velocity value says nothing about whether the speed is changing.
BIt is moving to the left with a speed of −4 m/s. A student who gives the magnitude the component's sign picks this. Speed is a magnitude and is never negative: the cart moves to the left at a speed of 4 m/s.
CIt is located 4 m to the left of the origin. A student who reads the velocity as a position picks this. vx = −4 m/s says which way and how fast the cart moves, not where it is; the cart could be anywhere on the track.
DIt is moving to the left with a speed of 4 m/s.Correct The minus sign gives the direction: the cart moves in the negative direction, to the left. The magnitude, 4 m/s, is its speed.
Working The sign gives the direction (negative: to the left); the magnitude, 4 m/s, is the speed.
A cart moves along a straight track, with the positive direction to the right. At one instant its acceleration is ax = −2 m/s². A student claims: "The cart must be slowing down." Which statement correctly evaluates the claim?
Answer and reasoning
AIt is justified: a negative acceleration means that the cart's speed is decreasing with time. A student who reads a negative acceleration as slowing down picks this. The minus sign gives the direction of the acceleration. If the cart is moving to the left, the same acceleration makes it speed up.
BIt is not justified: the minus sign tells only that the acceleration points to the left.Correct In one dimension the sign of a component gives its direction and nothing else: ax = −2 m/s² means the acceleration points to the left. The cart slows down if it is moving to the right but speeds up if it is moving to the left, and its velocity is not given.
CIt is not justified: the minus sign shows instead that the cart must be moving to the left. A student who thinks acceleration points in the direction of motion picks this. The sign of ax says which way the velocity is changing, not which way the cart moves; the cart could be moving either way.
DIt is not justified: an acceleration cannot be negative, so it is really +2 m/s². A student who treats a component as a magnitude picks this. The magnitude of the acceleration, 2 m/s², cannot be negative, but its component ax can: the minus sign shows that it points to the left.
Working The sign of ax gives only the direction of the acceleration (to the left). With vx > 0 the cart slows down; with vx < 0 it speeds up. The velocity is not given, so the claim does not follow.
A hiker walks 5 m east and then 9 m west along a straight path. Student 1 takes east as the positive direction; Student 2 takes west as the positive direction. Each finds the hiker's resultant displacement as a signed component. Which statement correctly compares their results?
Answer and reasoning
AOnly Student 1 can be right, as east has to be the positive direction. A student who thinks the positive direction is fixed picks this. Either direction may be chosen as positive; Student 2's +4 m, with west positive, describes the same displacement as Student 1's −4 m.
BStudent 1 gets −4 m and Student 2 gets +4 m, and both results mean 4 m west.Correct Student 1: +5 m + (−9 m) = −4 m. Student 2: −5 m + (+9 m) = +4 m. The signs differ because the axes differ, but in both coordinate systems the result means 4 m to the west, the same physical displacement.
CBoth get +4 m, by subtracting the smaller distance from the larger. A student who subtracts magnitudes and drops the sign picks this. With east positive the resultant points west and so is negative, −4 m; only Student 2's axis gives +4 m.
DBoth get 14 m, since the two walks simply add to give the resultant. A student who adds the magnitudes of opposite vectors picks this. 14 m is the distance walked. The walks are in opposite directions, so in either coordinate system their components have opposite signs and partly cancel.
Working East positive: (+5 m) + (−9 m) = −4 m. West positive: (−5 m) + (+9 m) = +4 m. Both mean 4 m west of the starting point.
Compiled from the AP Physics 1 Course and Exam Description (effective Fall 2024, 2026 reissue) and our question bank · Specialist review in progress. How these pages are made · Free, no account
