Position, Velocity, and Acceleration Using Integrals - AP Calculus AB

1. What is the physical interpretation of $v(t) = 0$?

   Object is momentarily at rest. Zero velocity means no instantaneous motion.

2. What is the integral of zero acceleration over time?

   Constant velocity. Integrating zero gives constant velocity.

3. If $v(t) = 0$ for all $t$, what can be said about $s(t)$?

   $s(t)$ is constant. Zero velocity means no change in position.

4. What is the algebraic expression for displacement?

   $\text{integral of } v(t) \text{ from } t_1 \text{ to } t_2$. Definite integral of velocity over time interval.

5. If $s(t) = 4t - \text{ln}(t)$, what is $v(t)$?

   $v(t) = 4 - \frac{1}{t}$. Differentiate using power and logarithmic rules.

6. What is the initial condition used for when integrating velocity to find position?

   To find the constant of integration. Determines the arbitrary constant $C$ from integration.

7. How do you find total distance traveled from a velocity function $v(t)$?

   Integrate $|v(t)|$ over the interval. Absolute value accounts for direction changes.

8. What does the area under a velocity-time graph represent?

   Displacement. Area under curve equals change in position.

9. What is the significance of the constant of integration when finding $s(t)$?

   It represents initial position. The constant $C$ represents position at $t = 0$.

10. What is the implication of $a(t) = 0$ over an interval?

    Velocity is constant. Zero acceleration means no change in velocity.

11. What is the derivative of position with respect to time?

    Velocity. Rate of change of position is velocity by definition.

12. If $v(t) = t^3 - 6t^2 + 9t$, find $a(t)$.

    $a(t) = 3t^2 - 12t + 9$. Take the derivative of velocity using power rule.

13. What does the integral of acceleration with respect to time represent?

    Change in velocity. Integration gives net change; $\int a(t) dt = \Delta v$.

14. What is the derivative of velocity with respect to time?

    Acceleration. Rate of change of velocity is acceleration by definition.

15. If velocity is constant, what is acceleration?

    Zero. Derivative of constant velocity is zero.

16. How do you find average velocity over an interval $[a, b]$?

    Average velocity = $\frac{s(b) - s(a)}{b - a}$. Displacement divided by time elapsed.

17. Given $v(t) = 4t - 7$, find the acceleration $a(t)$.

    $a(t) = 4$. Take the derivative of velocity function.

18. If $v(t) = 2t + 1$, what is the integral to find total distance?

    Integrate $|2t + 1|$ over the interval. Use absolute value when velocity changes sign.

19. What does the integral of velocity with respect to time represent?

    Position. Integration reverses differentiation; $\int v(t) dt = s(t)$.

20. Given $a(t) = 6t$, find the velocity function $v(t)$ if $v(0) = 3$.

    $v(t) = 3t^2 + 3$. Integrate $a(t) = 6t$ and apply $v(0) = 3$.

21. If $a(t) = 8 - 2t$, find $v(t)$ given $v(0) = 0$.

    $v(t) = 8t - t^2$. Integrate acceleration and apply zero initial velocity.

22. If $a(t) = 0$, what can be said about $v(t)$?

    $v(t)$ is a constant function. Zero acceleration implies constant velocity function.

23. Given $s(t) = 2t^3 - 5t^2 + 3t + 1$, find $v(t)$.

    $v(t) = 6t^2 - 10t + 3$. Take the derivative of position function.

24. If $v(t) = 3t$, find $s(t)$ given $s(0) = 0$.

    $s(t) = \frac{3t^2}{2}$. Integrate velocity and apply zero initial position.

25. If $s(t) = t^2 - 3t + 2$, what is $v(t)$?

    $v(t) = 2t - 3$. Differentiate position function using power rule.

26. For $a(t) = 4$, find $v(t)$ and $s(t)$ given $v(0) = 2$ and $s(0) = 3$.

    $v(t) = 4t + 2$, $s(t) = 2t^2 + 2t + 3$. Integrate twice with given initial conditions.

27. If $s(t) = \text{ln}(t)$, what is $v(t)$?

    $v(t) = \frac{1}{t}$. Differentiate logarithmic function.

28. How do you determine if an object is speeding up or slowing down?

    Check if $v(t)$ and $a(t)$ have the same sign. Same sign means speeding up, opposite means slowing down.

29. If $v(t) = t^2 + 3t + 2$, what is the position function $s(t)$ if $s(0) = 4$?

    $s(t) = \frac{t^3}{3} + \frac{3t^2}{2} + 2t + 4$. Integrate $v(t)$ and use initial condition $s(0) = 4$.

30. Given $v(t) = 5t - 4$, find the integral to determine position.

    Integrate $v(t)$ to get $s(t) = \frac{5t^2}{2} - 4t + C$. Apply antiderivative rules for polynomial functions.

31. Given $v(t) = 6t - 3t^2$, find $s(t)$ if $s(0) = 5$.

    $s(t) = 3t^2 - t^3 + 5$. Integrate velocity and use initial condition.

32. What is the formula for displacement from time $t_1$ to $t_2$?

    $s(t_2) - s(t_1) = \text{integral of } v(t) \text{ from } t_1 \text{ to } t_2$. Net change in position equals integral of velocity.

33. For $v(t) = 2t^3 + 3$, find the acceleration $a(t)$.

    $a(t) = 6t^2$. Differentiate velocity using power rule.

34. What is the relationship between acceleration and concavity in position function?

    Positive acceleration means upward concavity. Second derivative determines concavity of position graph.

35. If $a(t) = 3$, find $v(t)$ given $v(0) = 5$.

    $v(t) = 3t + 5$. Integrate constant acceleration and apply initial condition.

36. What condition must be checked when integrating to determine total distance traveled?

    Check for sign changes in $v(t)$. Must split integral where velocity changes sign.

37. Given $s(t) = 4t^2 - 2t + 1$, find $v(t)$ and $a(t)$.

    $v(t) = 8t - 2$, $a(t) = 8$. Differentiate position once for $v(t)$, twice for $a(t)$.

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