Vectors and Motion in Two Dimensions - AP Physics 1

1. What is the horizontal component of velocity for a projectile at launch speed $v$ and angle $\theta$?

   $v \cos \theta$. Horizontal component stays constant.

2. Identify the unit vector in the direction of $(6, 8)$.

   $(0.6, 0.8)$. Divide by magnitude $\sqrt{6^2 + 8^2} = 10$.

3. What is the effect of air resistance on projectile motion?

   It decreases range and height. Air drag opposes motion.

4. State the formula for centripetal acceleration.

   $a_c = \frac{v^2}{r}$. Acceleration toward center.

5. What variable represents angular velocity?

   $\omega$. Greek omega represents rotation rate.

6. Identify the unit of angular acceleration.

   $\text{rad/s}^2$. Angular acceleration measured in radians.

7. What is the range formula for a projectile launched at $v$ and angle $\theta$?

   $R = \frac{v^2 \sin 2\theta}{g}$. Uses $\sin(2\theta)$ identity.

8. Which factor remains constant in a projectile's horizontal motion?

   Horizontal velocity. No horizontal acceleration exists.

9. What is the angle between vectors $(1, 0)$ and $(0, 1)$?

   $90^{\circ}$. Unit vectors along axes are perpendicular.

10. Define the term 'scalar quantity'.

    A quantity with magnitude only. No directional information included.

11. What is the cross product of vectors used to find?

    Vector perpendicular to both vectors. Cross product creates perpendicular vector.

12. What is the formula for the gravitational potential energy of an object?

    $U = mgh$. Height-dependent gravitational energy.

13. Identify the velocity of a projectile at its highest point.

    Zero vertical velocity. Instantaneously stops rising.

14. What is the scalar product used for in vector calculations?

    Calculating work done. Dot product gives work value.

15. What is the two-dimensional motion with constant acceleration called?

    Projectile motion. Motion under gravity in two dimensions.

16. State the component form of vector $\textbf{A}$ with magnitude $5$ at $30^{\circ}$.

    $(4.33, 2.5)$. Use $x = 5\cos(30°)$, $y = 5\sin(30°)$.

17. State the formula for the dot product of vectors $(a_1, b_1)$ and $(a_2, b_2)$.

    $a_1 a_2 + b_1 b_2$. Multiply corresponding components and sum.

18. What is the dot product of vectors $(2, 3)$ and $(4, -1)$?

    $5$. $(2)(4) + (3)(-1) = 8 - 3 = 5$.

19. State the formula for the maximum height of a projectile.

    $h = \frac{v^2 \sin^2 \theta}{2g}$. From energy or kinematics.

20. What is the projection of vector $\textbf{A}$ onto vector $\textbf{B}$?

    $\frac{\textbf{A} \cdot \textbf{B}}{||\textbf{B}||}$. Component of A along B direction.

21. What is the formula for the velocity of an object in circular motion?

    $v = r \omega$. Links linear and angular motion.

22. What is the direction of centripetal force in circular motion?

    Toward the center of the circle. Keeps object in circular path.

23. What is the relationship between period and frequency?

    $T = \frac{1}{f}$. Period and frequency are reciprocals.

24. What is the resultant vector of $(3, 4)$ and $(1, 2)$?

    $(4, 6)$. Add corresponding components.

25. State the relationship between linear speed and angular speed.

    $v = r \omega$. Fundamental circular motion relationship.

26. Identify the vertical component of velocity for a projectile at launch speed $v$ and angle $\theta$.

    $v \sin \theta$. Initial vertical velocity component.

27. What is the formula for the cross product magnitude of $(a_1, b_1)$ and $(a_2, b_2)$?

    $|a_1 b_2 - a_2 b_1|$. 2D cross product magnitude formula.

28. What is the formula for work done by a force in the direction of displacement?

    $W = Fd \cos \theta$. Force component in displacement direction.

29. Which operation combines two vectors to form a resultant vector?

    Vector addition. Forms resultant from two vectors.

30. Find the vector sum of $(1, 2, 3)$ and $(4, 5, 6)$.

    $(5, 7, 9)$. Add components: $(1+4, 2+5, 3+6)$.

31. What does the magnitude of the cross product represent?

    Area of the parallelogram formed. Geometric interpretation of cross product.

32. State the formula for the magnitude of a vector $(x, y)$.

    $\text{Magnitude} = \sqrt{x^2 + y^2}$. Pythagorean theorem for vector length.

33. What is the definition of a vector quantity?

    A quantity with both magnitude and direction. Combines size and direction info.

34. What is the acceleration due to gravity in projectile motion?

    $9.8 \text{ m/s}^2$ downward. Standard gravity near Earth's surface.

35. State the formula for angular displacement.

    $\theta = \omega t$. Angle equals rate times time.

36. What condition indicates two vectors are perpendicular?

    Dot product equals zero. Orthogonal vectors have zero dot product.

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