Solved examples of breaking down velocity vectors

📚 Quick Study Guide

• 📏 Vector Components: Any velocity vector $\vec{v}$ can be broken down into horizontal ($v_x$) and vertical ($v_y$) components.
• 📐 Trigonometry: Use trigonometry to find the components. If $\theta$ is the angle with respect to the horizontal:
  • $v_x = v \cos(\theta)$
  • $v_y = v \sin(\theta)$
• ➕ Pythagorean Theorem: The magnitude of the velocity vector can be found using the Pythagorean theorem: $v = \sqrt{v_x^2 + v_y^2}$.
• 🧭 Direction: The direction (angle) can be found using the inverse tangent: $\theta = \arctan(\frac{v_y}{v_x})$.
• ✍️ Sign Conventions: Pay attention to the signs of the components. $v_x$ is positive if pointing to the right, negative if pointing to the left. $v_y$ is positive if pointing upwards, negative if pointing downwards.

Practice Quiz

1. A ball is thrown with an initial velocity of 20 m/s at an angle of 30° above the horizontal. What is the horizontal component of the velocity?
   1. 10 m/s
   2. 17.3 m/s
   3. 20 m/s
   4. 0 m/s
2. A plane is flying at a velocity of 100 m/s at an angle of 60° above the horizontal. What is the vertical component of the velocity?
   1. 50 m/s
   2. 86.6 m/s
   3. 100 m/s
   4. 0 m/s
3. A car is moving with a velocity that has a horizontal component of 15 m/s and a vertical component of 8 m/s. What is the magnitude of the velocity?
   1. 7 m/s
   2. 17 m/s
   3. 23 m/s
   4. 113 m/s
4. A projectile is launched with a velocity of 30 m/s at an angle of 45° above the horizontal. What are the horizontal and vertical components of the velocity?
   1. $v_x$ = 15 m/s, $v_y$ = 15 m/s
   2. $v_x$ = 21.2 m/s, $v_y$ = 21.2 m/s
   3. $v_x$ = 30 m/s, $v_y$ = 0 m/s
   4. $v_x$ = 0 m/s, $v_y$ = 30 m/s
5. A boat is traveling at a velocity of 8 m/s east and a current is pushing it at 6 m/s north. What is the magnitude of the resultant velocity?
   1. 2 m/s
   2. 10 m/s
   3. 14 m/s
   4. 48 m/s
6. A hiker walks 5 km east and 3 km north. What is the angle of the hiker's displacement vector with respect to the east?
   1. 30.96°
   2. 59.04°
   3. 36.87°
   4. 53.13°
7. A rocket is fired with a velocity of 50 m/s at an angle of 20° above the horizontal. What is the vertical component of its velocity?
   1. 46.98 m/s
   2. 17.10 m/s
   3. 50 m/s
   4. 0 m/s