Two-Dimensional Motion and Vectors Vector Operations

1. Two-Dimensional Motion and VectorsVector Operations Chapter 3: Section 2

2. Learning Targets • Identify appropriate coordinate systems for solving problems with vectors • Apply the Pythagorean theorem and tangent functions to calculate the magnitude and direction of a resultant vector • Resolve vectors into components using the sine and cosine functions • Add vectors that are not perpendicular P3.2A, P3.2C

3. Determining Resultant Magnitude and Direction • Previously, a graphical method was used to determine the magnitude and direction of a resultant vector • For vectors that are perpendicular to one another, a simpler method uses the Pythagorean theorem and the tangent function

4. The Pythagorean Theorem • The Pythagorean theorem can be used to find the magnitudeof the resultant (hypotenuse) if you know the magnitude of both the x and y components (length of hypotenuse)2= (length of one leg)2+ (length of other leg)2 Resultant

5. The Tangent Function • The inversetangent function can be used to find the direction of the resultant tangent of angle = opposite leg (∆y)/adjacent leg (∆x) • While the formula above gives the ratio of opposite to adjacent, the inverse of the tangent function (tan-1), must be used to find the angle

6. Resolving Vectors Into Components • The horizontal and vertical vectors that add up to give the actual displacement are called components • The x component is parallel to the x-axis while the y component is parallel to the y-axis

7. Solving for X and Y Components • To solve for the x and y components of a vector diagram, the sine and cosine functions can be used • To calculate vertical (y) component • Sine of angle = opposite leg (y)/hypotenuse • To calculate horizontal (x) component • Cosine of angle = adjacent leg (x) /hypotenuse • Remember the mnemonic device SOH-CAH-TOA

8. SOH-CAH-TOA

9. Example • A truck carrying a film crew must be driven at the correct velocity to enable the crew to film the underside of a biplane. The plane flies at 95 km/h at an angle of 20 degrees relative to the ground. At what speed must the film crew drive?

10. To find out the velocity that the truck must maintain to stay beneath the plane, we must know the horizontal component (vx)of the plane’s velocity SOHCAHTOA

11. To calculate vertical speed(y component) • Sine of an angle = opposite leg/hypotenuse • To calculate horizontal speed (x component) • Cosine of an angle = adjacent leg/hypotenuse

12. Adding Non-Perpendicular Vectors • Suppose that a plane initially travels at 5 km at an angle of 35° to the ground, then climbs at only 10° relative to the ground for 22 km. How can you determine the magnitude and direction for the vector denoting the total displacement of the plane?

13. Because the original displacement vectors do not form a right triangle, you cannot apply the tangent function or the Pythagorean theorem • Determining the magnitude and direction of the resultant can be achieved by resolving each of the plane’s displacement vectors into it x and y components • Then use the Pythagorean theorem and inverse tangent function