Understanding Vectors in Physics - Properties, Operations, and Applications

1. Physics Chapter 3 Vectors

2. Projectiles Launched at an Angle • The projectile has both an initial vertical component of velocity and a horizontal component of velocity • If the initial velocity vector makes and angle with the horizontal, the object’s motion must be resolved into its components---to do so use the sine and cosine functions to find the horizontal and vertical components of the initial velocity

3. Scalars and Vectors • Vectors indicate direction • Scalar – a physical quantity that has only a magnitude but no direction. Ex. Speed, volume, density, etc. – with appropriate units • Vector – a physical quantity that has both a magnitude and a direction. Ex. Displacement, velocity, acceleration and force

4. Vectors • Represented by symbols. In this book – vectors are boldface. Scalars are in italics • To keep track of vectors and their directions, use diagrams. Vectors are shown as arrows that point in the direction of the vector – length of arrow is related to magnitude • Can be added graphically – must have the same units • The answer found by adding 2 or more vectors = resultant or the vector sum

6. Properties of Vectors • Determining a resultant vector by drawing a vector from the tail of the first vector to the tip of the last vector = triangle method of addition • The magnitude of the resultant vector can be measured using a ruler, and the angle can be measured with a protractor • 2 or more vectors can be added in any order

9. Adding and Subtracting Vectors • To subtract a vector, add its opposite – a negative vector is defined as a vector with the same magnitude as the original vector but opposite in direction • Ex. 30 m/s west = - 30 m/s west or 30 m/s east. • Add a vector to its negative vector gives zero • When adding vectors in 2 dimensions, you can add a negative vector to a positive vector that does not point along the same line by using the triangle method of addition

10. Multiplying or dividing vectors by scalars results in vectors. Ex. “go twice as fast” turns into . “go twice as fast in the opposite direction”

11. Vector Operations • Movements to the right along the x axis and upward along the y axis = positive • Movement to the left along the x axis and downward along the y axis = negative • Diagramming the motion of an object in 2 dimensions employs vectors and the use of both the x and y axes simultaneously – the addition of another axis (from Ch. 2) helps describe 2 dimensional motion and simplifies analysis of motion in one dimension

12. Determining resultant magnitude and direction • There are drawbacks to graphically drawing resultant magnitude and direction  time consuming and accuracy of the answer depends on how carefully the diagram is drawn and measured • Simpler method  use Pythagorean Theorem and the tangent function

13. Pythagorean Theorem – can be applied to any right triangle • It can be applied to find the magnitude of the resultant displacement • It states that for any right triangle the square of the hypotenuse (side opposite to the right angle) equals the sum of the squares of the legs • (length of hypotenuse)2 = (length of 1 leg)2 + (length of the other leg)2 • c2 = a2 + b2 • or d2 = ∆x2 + ∆y2 • displacement2 = horizontal displacement2 + vertical displacement2

14. use the tangent function to find the direction of the resultant • you must know the direction of an object’s motion to completely describe the objects displacement • the tangent function can be used to find the angle Θ, this denotes the direction of the objects displacement • tan Θ = Opp/adj • tangent of angle = opposite leg/adjacent leg • to find the angle – use the inverse of the tangent function • Θ = tan -1 (opp/adj)

15. Resolving vectors into components • Components of a vector – the projections of a vector along the axes of a coordinate system. Ex. In the pyramid example – the horizontal and vertical parts that add up to give the object’s actual displacement = components • X component – parallel to x axis • Y component – parallel to y axis • Can be either positive or negative numbers

16. When a vector points along a single axis – with units – the second component of the vector is zero • When breaking a single vector into 2 components (resolving it into the components) an object’s motion can sometimes be described in terms of directions – north to south or east to west

17. Sine and cosine functions are defined in terms of the lengths of the sides of right triangles • sin Θ = opp/hyp • sine of angle = opposite leg/ hypotenuse • the leg opposite the 20° angle represents the y component – Vy – and describes the vertical speed of the airplane

19. the hypotenuse, plane is the resultant vector that describes the airplane’s total motion • cosine of an angle is the ration between the leg adjacent to that angle and the hypotenuse • cos Θ= adj/hyp • cosine of angle = adjacent leg/ hypotenuse • the leg adjacent to the 20° angle represents the x component – Vx – and describes the horizontal speed of the airplane • the x component equals the speed that the truck must maintain to stay beneath the plane

20. Adding vectors that are not perpendicular • if the original displacement vectors do not form a right triangle, it is not possible to directly apply the tangent function or the Pythagorean theorem when adding the original 2 vectors

21. to achieve the magnitude and the direction of the resultant – resolve each of the object’s displacement vectors into their x and y components then add the components along each axis together – the vector sums will be the 2 perpendicular components of the resultant • the magnitude of the resultant – use Pythagorean theorem • the direction – use the tangent function

22. Projectile Motion • When an object is propelled into the air in a direction other than straight up or down, the velocity, acceleration and displacement of the object do not all point in the same direction – this makes the vector forms of the equations difficult to solve. • Apply the technique of resolving vectors into components – then apply the simpler one-dimensional forms of the equations for each component – then recombine the components to determine the resultant

24. Figure 3-17 – the jumper’s velocity vector is resolved into its 2 component vectors which allows the motion to be analyzed using the kinematic equations applied to one direction at a time

25. Objects that are thrown or launched into the air and are subject to gravity are called projectiles • Ex. Softballs, footballs, the long jumper • The path of a projectile is a curve called a parabola • Projectiles don’t fall straight down because of its initial horizontal velocity – if an object has an initial horizontal velocity in any given interval, there will be horizontal motion throughout the flight of the projectile • Airplanes have an initial horizontal velocity but are not considered projectiles. Why?

26. (In this book the horizontal velocity of the projectile will be a constant). It would not be constant if we considered air resistance

27. Parabola – is a curve in which every point is the same difference from a fixed point or the focus • With air resistance, projectiles slow down as they collide with air particles – the true path of a projectile traveling through Earth’s atmosphere is not a parabola.

28. Projectile Motion is free fall with an initial horizontal velocity • The red ball is released at the same instant the yellow ball is launched horizontally. If air resistance is disregarded, both balls hit the ground at the same time

30. Relative Motion • Velocity measurements differ in different frames of reference • Observers using different frames of reference may measure different velocities for an object in motion • Ex. The two views of the airplane

31. Relative Velocity • There is no general equation to work relative velocity problems • It can be shown as the difference of the 2 vectors or relative velocity of one moving object to another is the difference between their velocities relative to some common reference point • Ex. Car moving at +80 km/h N and fast car moving at +90 km/hr N. Slow car gets the – vector because to the observer in the slow car, the Earth is moving South at a velocity of 80 km/h • Vfs = +90 km/h – 80km/h = +10 km/h