AP Calculus AB Quiz: Connecting Position Velocity And Acceleration

What this quiz covers

This quiz focuses on Connecting Position Velocity And Acceleration, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus AB.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Moving right and slowing down (correct answer)
2. Moving right and speeding up
3. Moving left and slowing down
4. Moving left and speeding up
5. At rest and slowing down

Explanation: This straight-line motion analysis uses graphical interpretation of velocity behavior. When the velocity graph is above the t-axis, $v > 0$, so the particle moves right. If the velocity graph is decreasing at $t = 1$, then $v'(1) < 0$, meaning acceleration is negative. With positive velocity and negative acceleration, they have opposite signs, so the particle is slowing down. Students might think positive velocity means speeding up, but negative acceleration opposes positive velocity. The key strategy is to determine if velocity and acceleration have the same sign (speeding up) or opposite signs (slowing down).

Question 2

1. Moving right and slowing down (correct answer)
2. Moving right and speeding up
3. Moving left and slowing down
4. Moving left and speeding up
5. At rest and slowing down

Explanation: This straight-line motion analysis covers motion over an interval with specific velocity and acceleration signs. Since $v(t) > 0$ on $(0,3)$, the object moves right throughout this interval. Given $a(t) < 0$ on $(0,3)$, the acceleration is negative, pointing left and opposing the rightward motion. When velocity and acceleration have opposite signs, the object slows down. Students might incorrectly choose "moving right and speeding up" by focusing only on the positive velocity. The correct motion analysis strategy is to check if velocity and acceleration have the same sign (speeding up) or opposite signs (slowing down).

Question 3

1. Moving left and slowing down
2. Moving right and speeding up (correct answer)
3. Moving right and slowing down
4. Moving left and speeding up
5. At rest and speeding up

Explanation: This straight-line motion analysis examines consistent positive velocity and acceleration over an interval. Since $v(t) > 0$ on $(0,2)$, the object moves right throughout this interval. Given $a(t) > 0$ on $(0,2)$, the acceleration also points right, in the same direction as the motion. When velocity and acceleration have the same sign (both positive), the object speeds up. Students rarely make errors with this case since both quantities point in the same direction. The key motion analysis strategy is that same signs of velocity and acceleration always mean speeding up.

Question 4

1. A change in direction (correct answer)
2. A maximum in speed
3. Acceleration must be zero
4. Position must be zero
5. Velocity is constant near $t=3$

Explanation: This straight-line motion analysis focuses on what happens when velocity changes sign at a specific time. Since $v(3) = 0$, the particle is momentarily at rest at $t = 3$. The fact that velocity changes from negative to positive means the particle changes from moving left to moving right. This represents a change in direction of motion. Students might think about speed maxima or zero acceleration, but the key insight is the direction change. The correct motion analysis strategy is to recognize that when velocity changes sign (passes through zero), the particle changes direction.

Question 5

1. Position is increasing (correct answer)
2. Position is decreasing
3. Speed is decreasing
4. Acceleration is positive
5. Velocity is constant

Explanation: This straight-line motion analysis examines what must be true when velocity is consistently positive over an interval. Since $v(t) > 0$ for all $t$ in $(1,4)$, the velocity is always positive on this interval. Positive velocity means the position function $s(t)$ is always increasing on $(1,4)$ because $s'(t) = v(t) > 0$. We cannot determine acceleration, speed changes, or specific values without more information. Students might think about acceleration or speed, but only position behavior is guaranteed. The key strategy is that positive velocity always means increasing position.

Question 6

1. Speed is increasing
2. Speed is decreasing (correct answer)
3. Speed is zero
4. Acceleration is negative
5. Velocity is positive

Explanation: This straight-line motion problem focuses on speed changes when velocity and acceleration have specific signs. Since $v(8) < 0$, the particle moves left at $t = 8$. Given $a(8) > 0$, the acceleration points right, opposing the leftward motion. When velocity and acceleration have opposite signs, the speed (magnitude of velocity) decreases. Students might think "speed is increasing" because acceleration is positive, but positive acceleration actually opposes negative velocity here. The correct motion analysis strategy is that speed decreases when velocity and acceleration have opposite signs.

Question 7

1. Moving left and slowing down (correct answer)
2. Moving right and speeding up
3. Moving left and speeding up
4. At rest and speeding up
5. Moving right and slowing down

Explanation: This problem requires analyzing straight-line motion using velocity and acceleration signs. Given $v(2) < 0$, the particle is moving left (negative direction) at $t = 2$. Since $a(2) > 0$, the acceleration is positive, pointing right. When velocity and acceleration have opposite signs, the particle is slowing down because acceleration opposes the direction of motion. Many students incorrectly choose "moving left and speeding up" by thinking positive acceleration always means speeding up. The key strategy is to check if velocity and acceleration have the same sign (speeding up) or opposite signs (slowing down).

Question 8

1. $s(t)$ has a local maximum at $t=1$
2. $s(t)$ has a local minimum at $t=1$ (correct answer)
3. $s(t)$ is decreasing at $t=1$
4. $s(t)$ is increasing at $t=1$
5. $s(t)$ is linear near $t=1$

Explanation: This straight-line motion analysis uses the second derivative test to analyze position function behavior. Since $s'(1) = 0$, the velocity is zero, creating a critical point for the position function at $t = 1$. Given $s''(1) > 0$, the second derivative is positive, indicating the position function is concave up at this critical point. By the second derivative test, this means $s(t)$ has a local minimum at $t = 1$. Students might confuse this with a maximum by misremembering the second derivative test. The key strategy is that $s'(c) = 0$ and $s''(c) > 0$ means a local minimum at $x = c$.

Question 9

1. Acceleration is negative (correct answer)
2. Acceleration is positive
3. Acceleration is zero
4. Acceleration equals velocity
5. Acceleration cannot be determined

Explanation: This straight-line motion problem connects velocity behavior to acceleration using derivative relationships. When velocity is positive and decreasing at $t = 5$, we have $v(5) > 0$ and $v'(5) < 0$. Since acceleration is the derivative of velocity, $a(5) = v'(5) < 0$, so acceleration is negative. Students might think positive velocity means positive acceleration, but decreasing velocity always means negative acceleration. The key motion analysis strategy is that acceleration equals the derivative of velocity, so decreasing velocity always indicates negative acceleration.

Question 10

1. Speed is increasing (correct answer)
2. Speed is decreasing
3. Speed is constant
4. Velocity is positive
5. Acceleration is positive

Explanation: This straight-line motion problem examines speed changes when both velocity and acceleration are negative. Since $v(2) < 0$, the particle moves left at $t = 2$. Given $a(2) < 0$, the acceleration also points left, in the same direction as the velocity. When velocity and acceleration have the same sign (both negative), the speed increases because acceleration enhances the leftward motion. Students might think negative acceleration means decreasing speed, but here both point left together. The correct motion analysis strategy is that same signs of velocity and acceleration mean increasing speed.

Question 11

1. Acceleration is positive (correct answer)
2. Acceleration is negative
3. Acceleration is zero
4. Acceleration equals speed
5. Acceleration must be undefined

Explanation: This straight-line motion analysis connects velocity changes to acceleration through derivative relationships. When velocity is negative and increasing at $t = 5$, we have $v(5) < 0$ and $v'(5) > 0$. Since acceleration equals the derivative of velocity, $a(5) = v'(5) > 0$, so acceleration is positive. Students might think negative velocity implies negative acceleration, but increasing velocity (even if negative) always means positive acceleration. The key strategy is that acceleration is the derivative of velocity, so increasing velocity always indicates positive acceleration regardless of velocity's sign.

Question 12

1. Moving left and slowing down
2. Moving right and slowing down (correct answer)
3. Moving right and speeding up
4. Moving left and speeding up
5. At rest and slowing down

Explanation: This straight-line motion analysis uses explicit notation showing that acceleration equals the derivative of velocity. Since $v(2) > 0$, the object moves right at $t = 2$. Given $a(2) = v'(2) < 0$, the acceleration is negative, pointing left and opposing the rightward motion. When velocity and acceleration have opposite signs, the object slows down. Students might think positive velocity always means speeding up, but negative acceleration reduces the rightward speed. The key motion analysis strategy is to check if velocity and acceleration have the same sign (speeding up) or opposite signs (slowing down).

Question 13

1. Moving left and speeding up
2. Moving left and slowing down (correct answer)
3. Moving right and slowing down
4. Moving right and speeding up
5. At rest and speeding up

Explanation: This straight-line motion problem explicitly shows acceleration as the derivative of velocity. Since $v(2) < 0$, the object moves left at $t = 2$. Given $a(2) = v'(2) > 0$, the acceleration is positive, pointing right and opposing the leftward motion. When velocity and acceleration have opposite signs, the object slows down. Students might think positive acceleration always means speeding up, but here it opposes the negative velocity. The correct motion analysis strategy is to determine if velocity and acceleration have the same sign (speeding up) or opposite signs (slowing down).

Question 14

1. Speed is increasing
2. Speed is decreasing (correct answer)
3. Speed is zero
4. Velocity is negative
5. Acceleration is positive

Explanation: This straight-line motion analysis examines speed changes when velocity and acceleration have opposite signs. Since $v(2) > 0$, the particle moves right at $t = 2$. Given $a(2) < 0$, the acceleration points left, opposing the rightward motion. When velocity and acceleration have opposite signs, the speed decreases because acceleration opposes the direction of motion. Students might think positive velocity always means increasing speed, but negative acceleration reduces rightward speed. The key motion analysis strategy is that opposite signs of velocity and acceleration always mean decreasing speed.

Question 15

1. Decreasing and concave up (correct answer)
2. Decreasing and concave down
3. Increasing and concave up
4. Increasing and concave down
5. Constant and concave down

Explanation: This straight-line motion analysis examines position function behavior with specific derivative signs. Since $s'(t) < 0$ on $(0,1)$, the position function is decreasing throughout this interval. Given $s''(t) > 0$ on $(0,1)$, the second derivative is positive, meaning the position function is concave up. A decreasing and concave up function falls while curving upward (like an upside-down smile). Students might confuse how concavity combines with increasing/decreasing behavior. The key strategy is that $s'(t) < 0$ means decreasing, and $s''(t) > 0$ means concave up.

Question 16

1. Increasing and concave down (correct answer)
2. Increasing and concave up
3. Decreasing and concave up
4. Decreasing and concave down
5. Constant and concave up

Explanation: This straight-line motion problem analyzes position function characteristics using derivative information. Since $s'(t) > 0$ on $(0,1)$, the position function is increasing throughout this interval. Given $s''(t) < 0$ on $(0,1)$, the second derivative is negative, meaning the position function is concave down. An increasing and concave down function rises while curving downward (like a regular smile). Students might confuse the signs of derivatives with function behavior. The key strategy is that $s'(t) > 0$ means increasing, and $s''(t) < 0$ means concave down.

Question 17

1. Increasing and concave up (correct answer)
2. Increasing and concave down
3. Decreasing and concave up
4. Decreasing and concave down
5. Constant and concave up

Explanation: This straight-line motion problem analyzes position function behavior using first and second derivatives. Since $s'(t) > 0$ on $(0,1)$, the position function is increasing throughout this interval. Given $s''(t) > 0$ on $(0,1)$, the second derivative is positive, meaning the position function is concave up. An increasing and concave up function curves upward as it rises. Students might confuse the relationship between derivatives and concavity. The key strategy is that $s'(t) > 0$ means increasing, and $s''(t) > 0$ means concave up.

Question 18

1. Moving left and slowing down (correct answer)
2. Moving left and speeding up
3. Moving right and slowing down
4. Moving right and speeding up
5. At rest and speeding up

Explanation: This straight-line motion problem describes velocity graphically and asks about motion characteristics. When the velocity graph is below the t-axis, $v < 0$, so the particle moves left. If the velocity graph is increasing at $t = 1$, then $v'(1) > 0$, meaning acceleration is positive. With negative velocity and positive acceleration, they have opposite signs, so the particle is slowing down. Students might think increasing velocity means speeding up, but increasing negative velocity means slowing down leftward motion. The key strategy is to check if velocity and acceleration have the same sign (speeding up) or opposite signs (slowing down).

Question 19

1. Moving right and speeding up
2. Moving left and speeding up
3. Moving right and slowing down
4. At rest and speeding up
5. Moving left and slowing down (correct answer)

Explanation: This problem requires analyzing straight-line motion by examining the signs of velocity and acceleration. At t=4, we calculate v(4)=(4-2)(4-5)=(2)(-1)=-2, which is negative, indicating leftward motion. The acceleration a(4)=2(4)-7=8-7=1 is positive, meaning the velocity is increasing (becoming less negative). Since v<0 and a>0, the particle is moving left while slowing down (speed decreasing as velocity approaches zero). Students often confuse "speeding up" with positive acceleration, but when velocity and acceleration have opposite signs, the object slows down. Remember: same signs mean speeding up, opposite signs mean slowing down.

Question 20

1. moving right and speeding up
2. moving left and speeding up
3. moving right and slowing down
4. moving left and slowing down (correct answer)
5. at rest

Explanation: This problem tests interpreting derivatives in the context of straight-line motion. Given s'(t)<0, the velocity is negative, indicating leftward motion. Since s''(t)>0, the acceleration is positive, meaning velocity is increasing (becoming less negative). With negative velocity and positive acceleration (opposite signs), the particle is slowing down - its speed is decreasing as it approaches a stop. The common error is thinking positive acceleration always means speeding up, but this is only true when velocity is also positive. Strategy: velocity sign determines direction, and matching signs of v and a mean speeding up while opposite signs mean slowing down.