Position, Velocity, and Acceleration Vectors Quiz with Answer Key

📚 Quick Study Guide

• 📍 Position Vector: Describes the location of an object in space relative to an origin.
• 🚗 Velocity Vector: Represents the rate of change of position with respect to time, including both speed and direction. Formula: $\vec{v} = \frac{d\vec{r}}{dt}$
• 💨 Acceleration Vector: Represents the rate of change of velocity with respect to time. Formula: $\vec{a} = \frac{d\vec{v}}{dt}$
• 📐 Vector Components: Vectors can be broken down into components along coordinate axes (e.g., x, y, z).
• ➕ Vector Addition/Subtraction: Use component-wise addition/subtraction.
• 💡 Constant Acceleration Equations: Useful when acceleration is constant. For example, $\vec{v} = \vec{v_0} + \vec{a}t$ and $\vec{r} = \vec{r_0} + \vec{v_0}t + \frac{1}{2}\vec{a}t^2$

🧪 Practice Quiz

1. A particle's position vector is given by $\vec{r} = (3t^2)\hat{i} - (6t)\hat{j}$ (meters), where t is in seconds. What is the velocity vector at t = 2s?
   1. (12$\hat{i}$ - 6$\hat{j}$) m/s
   2. (6$\hat{i}$ - 6$\hat{j}$) m/s
   3. (3$\hat{i}$ - 6$\hat{j}$) m/s
   4. (12$\hat{i}$ + 6$\hat{j}$) m/s
2. An object moves with a constant acceleration of $\vec{a} = (2\hat{i} + 3\hat{j})$ m/s². If its initial velocity is $\vec{v_0} = (\hat{i} - \hat{j})$ m/s, what is its velocity after 3 seconds?
   1. (7$\hat{i}$ + 8$\hat{j}$) m/s
   2. (6$\hat{i}$ + 9$\hat{j}$) m/s
   3. (7$\hat{i}$ + 9$\hat{j}$) m/s
   4. (5$\hat{i}$ + 8$\hat{j}$) m/s
3. A car accelerates from rest with a constant acceleration of $\vec{a} = (4\hat{i})$ m/s². What is its displacement vector after 5 seconds?
   1. (50$\hat{i}$) m
   2. (10$\hat{i}$) m
   3. (20$\hat{i}$) m
   4. (100$\hat{i}$) m
4. A projectile is launched with an initial velocity of $\vec{v_0} = (10\hat{i} + 20\hat{j})$ m/s. Assuming no air resistance, what is the acceleration vector during its flight?
   1. (0$\hat{i}$ - 9.8$\hat{j}$) m/s²
   2. (10$\hat{i}$ - 9.8$\hat{j}$) m/s²
   3. (0$\hat{i}$ + 9.8$\hat{j}$) m/s²
   4. (9.8$\hat{i}$ + 0$\hat{j}$) m/s²
5. If the position vector of an object is given by $\vec{r}(t) = (2t^3 - t)\hat{i} + (3t^2)\hat{j}$, what is the acceleration vector at t = 1 second?
   1. (12$\hat{i}$ + 6$\hat{j}$) m/s²
   2. (6$\hat{i}$ + 6$\hat{j}$) m/s²
   3. (12$\hat{i}$ + 3$\hat{j}$) m/s²
   4. (3$\hat{i}$ + 6$\hat{j}$) m/s²
6. An object’s velocity vector changes from $\vec{v_1} = (5\hat{i} - 2\hat{j})$ m/s to $\vec{v_2} = (2\hat{i} + 3\hat{j})$ m/s in 2 seconds. Calculate the average acceleration vector.
   1. (-1.5$\hat{i}$ + 2.5$\hat{j}$) m/s²
   2. (3.5$\hat{i}$ - 0.5$\hat{j}$) m/s²
   3. (-3$\hat{i}$ + 5$\hat{j}$) m/s²
   4. (7$\hat{i}$ - 1$\hat{j}$) m/s²
7. An object is moving with a velocity vector $\vec{v}(t) = (4t\hat{i} + 3\hat{j})$ m/s. What is the magnitude of its acceleration vector?
   1. 4 m/s²
   2. 3 m/s²
   3. 5 m/s²
   4. 7 m/s²