Motion in Two or Three Dimensions - AP Physics C Mechanics

1. State the formula for range of a projectile launched at angle $\theta$.

   $R = \frac{v_0^2 \text{sin}(2\theta)}{g}$. Maximum range occurs at 45° launch angle using trigonometric identity.

2. What is the relationship between centripetal force and mass?

   $F_c = ma_c$. Newton's second law applied to centripetal motion.

3. What is the formula for instantaneous velocity in vector form?

   $\textbf{v} = \frac{d\textbf{r}}{dt}$. Time derivative of position vector gives velocity vector.

4. Find the time of flight for a projectile launched horizontally from 20 m.

   $t = 2 \text{ s}$. Using $h = \frac{1}{2}gt^2$: $t = \sqrt{\frac{2h}{g}} = 2$ s.

5. What is the equation for projectile motion in the horizontal direction?

   $x = v_{0x}t + \frac{1}{2}a_xt^2$. Kinematic equation for horizontal motion with constant acceleration.

6. Calculate the magnitude of velocity given $v_x = 3$ m/s and $v_y = 4$ m/s.

   $v = 5 \text{ m/s}$. Pythagorean theorem: $\sqrt{3^2 + 4^2} = 5$ m/s.

7. Find the angular velocity for $T = 4$ s.

   $\omega = \frac{\text{π}}{2} \text{ rad/s}$. Angular velocity equals $\frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2}$ rad/s.

8. Calculate the centripetal force for $m = 2$ kg, $v = 3$ m/s, $r = 1$ m.

   $F_c = 18 \text{ N}$. Using $F_c = \frac{mv^2}{r} = \frac{2 \times 9}{1} = 18$ N.

9. What is the velocity vector at the peak of a projectile's trajectory?

   $v = v_{0x}\textbf{i}$. At peak, vertical velocity is zero, only horizontal remains.

10. What is the centripetal acceleration formula for circular motion?

    $a_c = \frac{v^2}{r}$. Acceleration directed toward center equals velocity squared over radius.

11. What is the formula for the horizontal component of initial velocity $v_0$?

    $v_{0x} = v_0 \text{cos}(\theta)$. Horizontal component using trigonometric cosine function.

12. What is the formula for the displacement vector in two dimensions?

    $\textbf{r} = x\textbf{i} + y\textbf{j}$. Position vector using unit vectors $\textbf{i}$ and $\textbf{j}$ for x and y components.

13. Identify the formula relating angular displacement to time.

    $\theta = \omega t$. Angular displacement equals angular velocity times time.

14. What is the formula for angular velocity $\omega$?

    $\omega = \frac{2\pi}{T}$. Angular velocity is $2\pi$ radians per period $T$.

15. Identify the unit vector notation for three-dimensional space.

    $\textbf{r} = x\textbf{i} + y\textbf{j} + z\textbf{k}$. Adds $\textbf{k}$ unit vector for the z-component in three dimensions.

16. Find the height an object reaches if launched vertically at 15 m/s.

    $h = 11.5 \text{ m}$. Using $h = \frac{v_0^2}{2g}$ for maximum height in vertical motion.

17. State the relationship between linear and angular velocity.

    $v = r\omega$. Linear velocity equals radius times angular velocity.

18. Calculate the maximum height reached by a projectile launched at 10 m/s, 30°.

    $h_{\text{max}} = 1.3 \text{ m}$. Using $h = \frac{v_0^2\sin^2\theta}{2g}$ with given values.

19. Identify the direction of acceleration for an object in uniform circular motion.

    Toward the center of the circle. Centripetal acceleration always points toward the rotation center.

20. State the formula for acceleration in vector form.

    $\textbf{a} = \frac{d\textbf{v}}{dt}$. Time derivative of velocity vector gives acceleration vector.

21. What is the formula for relative velocity in two dimensions?

    $\textbf{v}_{\text{rel}} = \textbf{v}_A - \textbf{v}_B$ . Velocity of object A relative to object B using vector subtraction.

22. What is the equation for projectile motion in the vertical direction?

    $y = v_{0y}t + \frac{1}{2}a_yt^2$. Kinematic equation for vertical motion with gravitational acceleration.

23. Calculate the horizontal distance given $v_{0x} = 5$ m/s, $t = 3$ s.

    $x = 15 \text{ m}$. Horizontal displacement with constant velocity: $x = v_x \times t$.

24. What is the maximum range for a projectile launched at 10 m/s?

    $R_{\text{max}} = 10.2 \text{ m}$. Maximum range occurs at 45° launch angle.

25. State the formula for average velocity in two dimensions.

    $\textbf{v}_{\text{avg}} = \frac{\textbf{r}_2 - \textbf{r}_1}{t_2 - t_1}$. Change in position divided by change in time for vector quantities.

26. Determine the resultant velocity given $v_x = 6$ m/s, $v_y = 8$ m/s.

    $v = 10 \text{ m/s}$. Pythagorean theorem: $\sqrt{6^2 + 8^2} = 10$ m/s.

27. Calculate the vertical velocity after 2 s for $v_{0y} = 0$, $a_y = -9.8$ m/s².

    $v_y = -19.6 \text{ m/s}$. Using $v_y = v_{0y} + a_yt = 0 + (-9.8)(2) = -19.6$ m/s.

28. Identify the gravitational acceleration constant near Earth's surface.

    $g = 9.8 \text{ m/s}^2$. Standard value for Earth's gravitational acceleration near surface.

29. Determine the horizontal range of a projectile launched at 10 m/s, 45°.

    $R = 10.2 \text{ m}$. Range formula with $v_0 = 10$ m/s and $\theta = 45°$.

30. What is the formula for the vertical component of initial velocity $v_0$?

    $v_{0y} = v_0 \text{sin}(\theta)$. Vertical component using trigonometric sine function.

31. How is the period $T$ of uniform circular motion related to velocity?

    $v = \frac{2\text{π}r}{T}$. Circumference divided by period gives tangential velocity in circular motion.

32. State the formula for the time of flight for a projectile launched at angle.

    $t_{\text{flight}} = \frac{2v_0 \sin(\theta)}{g}$. Time for projectile to return to launch height.

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