Vectors & 2-Dimensional Motion

Vectors &2-Dimensional Motion

Vectors & 2-D Motion • Adding vectors graphically • Vector Components • Adding vectors numerically • Relative velocity problems • Projectiles

Welcome to VectorLandall you Scalarians! • Adding vectors graphically • Head to tail addition with ruler & protractor. • Vector Components • Use trigonometry. • Adding vectors Numerically • Resolve all vectors into components using trig’. • Relative velocity problems • Boats relative to water…planes relative to air, etc. • Projectiles • Kinematics in 2-dimensions: constant horizontal velocity, constant vertical acceleration

Lots of in-class practice doing trigonometry • Master right-triangle trigonometry • SOH – CAH - TOA • Pythagorean theorem: a2 + b2 = c2 • Remember “head-to-tail” addition of vectors • Vector components add head-to-tail giving the original vector. • Please see Mr. Jones for outside help if you are struggling with trigonometry.

There are two kinds of quantities… • Vectors are quantities that have both magnitude and direction, such as • position & displacement • velocity • acceleration • Scalars are quantities that have magnitude only, such as • distance • speed • time • mass

Books usually write vector names in bold. You would write the vector name with an arrow on top. R R label R head tail Notating vectors: • This is how you notate a vector… • This is how you draw a vector… 

 x A Direction of Vectors • Vector direction is the direction of the arrow, given by an angle. • Draw a vector that has an angle that is between 0o and 90o.

y Quadrant II 90o <  < 180o Quadrant I 0 <  < 90o x     Quadrant III 180o <  < 270o Quadrant IV 270o <  < 360o Vector angle ranges

Between 180o and 270o x  B Direction of Vectors • What angle range would this vector have? • What would be the exact angle, and how would you determine it?

Magnitude of Vectors • The best way to determine the magnitude of a vector is to measure its length. • The length of the vector is proportional to the magnitude (or size) of the quantity it represents.

Equal Vectors • Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity). • Draw several equal vectors.

A -A Inverse Vectors • Inverse vectors have the same length, but opposite direction. • Draw a set of inverse vectors.

Warm-Up Problem Suppose you take the following trip, starting from your house. You go: 15 mi south, then 24 mi east, then 6 mi north, then 8 mi east, then 17 mi south, and finally 12 mi west. What is your total displacement, d from home? A B

Graphical Addition of Vectors • Add vectors A and B graphically by drawing them together in a head to tail arrangement. • Draw vector A first, and then draw vector B such that its tail is on the head of vector A. • Then draw the sum, or resultant vector, by drawing a vector from the tail of A to the head of B. • Measure the magnitude and direction of the resultant vector.

B A R Practice Graphical Addition A + B = R R is called the resultant vector!

The Resultant and the Equilibrant • The sum of two or more vectors is called the resultant vector. • The resultant vector can replace the vectors from which it is derived. • The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant.

B R A -R The Equilibrant Vector A + B = R The vector-R is called the equilibrant. If you add R and -R you get a null (or zero) vector.

Graphical Subtraction of Vectors • Subtract vectors A and B graphically by adding vector A with the inverse of vector B (-B). • First draw vector A, then draw -B such that its tail is on the head of vector A. • The difference is the vector drawn from the tail of vector A to the head of -B.

B -B C A Practice Graphical Subtraction A - B = C

Practice Problem • Vector A points in the +x direction and has a magnitude of 75 m. Vector B has a magnitude of 30 m and has a direction of 30o relative to the x axis. Vector C has a magnitude of 50 m and points in a direction of -60o relative to the x axis. • Find A + B • Find A + B + C • Find A – B.

Vector Addition Practice Add the following vectors graphically. Construct the vectors head-to-tail and measure the resultant with ruler and protractor. 1. A = 7.00 cm @ 140o 2. C = 50.0 km @ 325o B = 6.00 cm @ 270oD = 100.0 km @ 70o 3. E = 3.70 m/s @ 310.0o 4. G = 16 m/s2 @ 45o F = 7.90 m/s @ 90.0oH = 3.0 m/s2 @ 45o 5.I = 105 m @ 140.0o 6. K = 40.0 m/s @ 180o J = 35.0 m @ 320.0oL = 40.0 m/s @ 0o

What if we add three vectors head-to-tail? 7. M = 5.70 cm @ 320.0o N = 4.90 cm @ 90.0o O = 6.00 cm @ 210.0o Challenge: subtract these vectors… 8.P = 10.0 m @ 90.0o Q = 7.5 m @ 0.0o P – Q = ? Hint: P – Q = P + (-Q) = ?

Vector Components • The components of a vector are the two perpendicular vectors whose combined effect is the same as the original vector. • Head-to-tail, the components add up to make the original vector. • To indicate the direction of x & y components, a “+” or “–” is used (not a direction angle). • A vector’s components are determined by visualizing the 2 vectors in the x and y directions which will add up to the original vector.

Magnitude & Direction x & y Components

A  x Ax Vectors: x-component • The x-component of a vector is the “shadow” it casts on the x-axis. • It is calculated by Ax = A cos  • Draw the x-component of the vector below.

A  Ay x Vectors: y-component • The y-component of a vector is the “shadow” it casts on the y-axis. • It is calculated by Ay = A sin  • Draw the y-component of the vector below. y

Vectors: angle • The angle a vector makes with the x-axis can be determined by the components. • It is calculated by the inverse tangent function •  = tan-1 (Ry/Rx) y Ry  x Rx

Vectors: magnitude • The magnitude of a vector can be determined by the components. • It is calculated using the Pythagorean Theorem. • R2 = Rx2 + Ry2 y R Ry x Rx

Magnitude & Direction x & y Components

For each vector, sketch and calculate its x & y components. A = 8.50 m @ 310o B = 17.2 m/s @ 165o C = 65 cm @ 270o D = 5.00 m @ 215o

For each set of components, sketch and calculate the vector’s magnitude & direction. Ex = 2.0 m/s2 Ey = 9.8 m/s2 Fx = 26.0 m/s Fy = 0 Gx = 4.00 km Gy = 2.00 km

Practice Problem • You are driving up a long straight inclined road. After 1.5 miles you notice that signs along the roadside indicate that your elevation has increased by 520 feet. • What is the angle of the road above the horizontal? • How far do you have to drive to gain an additional 150 feet of elevation?

Practice Problem • Find the x- and y-components of a position vector r of magnitude 75 meters if its angle relative to the x-axis is • 25o • 95o

Component Addition of Vectors • Resolve each vector into its x- and y-components. Ax = Acos Ay = Asin Bx = Bcos By = Bsin Cx = Ccos Cy = Csin etc. • Add the x-components (Ax, Bx, etc.) together to get Rx and the y-components (Ay, By, etc.) to get Ry.

Component Addition of Vectors • Calculate the magnitude of the resultant with the Pythagorean Theorem (R = Rx2 + Ry2). • Determine the angle with the equation  = tan-1 Ry/Rx.

Practice Problem • In a daily prowl through the neighborhood, a cat makes a displacement of 120 m due north, followed by a displacement of 72 m due west. Find the magnitude and displacement required if the cat is to return home.

Practice Problem • If the cat in the previous problem takes 45 minutes to complete the first displacement and 17 minutes to complete the second displacement, what is the magnitude and direction of its average velocity during this 62-minute period of time?

Practice Problem • Vector A points in the +x direction and has a magnitude of 75 m. Vector B has a magnitude of 30 m and has a direction of 30o relative to the x axis. Vector C has a magnitude of 50 m and points in a direction of -60o relative to the x axis. • Find A + B • Find A + B + C • Find A – B.

Relative Motion • Relative motion problems are difficult to do unless one applies vector addition concepts. • Define a vector for a swimmer’s velocity relative to the water, and another vector for the velocity of the water relative to the ground. Adding those two vectors will give you the velocity of the swimmer relative to the ground.

Vs Vw Vw Vt = Vs +Vw Relative Motion

Practice Problem • You are paddling a canoe in a river that is flowing at 4.0 mph east. You are capable of paddling at 5.0 mph. • If you paddle east, what is your velocity relative to the shore? • If you paddle west, what is your velocity relative to the shore? • You want to paddle straight across the river, from the south to the north.At what angle to you aim your boat relative to the shore? Assume east is 0o.

Introduction to 2D Motion

2-Dimensional Motion • Definition: motion that occurs with both x and y components. • Example: • Playing pool . • Throwing a ball to another person. • Each dimension of the motion can obey different equations of motion.

Solving 2-D Problems • Resolve all vectors into components • x-component • y-component • Work the problem as two separate one-dimensional problems. • Each dimension can obey different equations of motion. • Re-combine the results for the two components at the end of the problem.

Sample Problem • You run in a straight line at a speed of 5.0 m/s in a direction that is 40o south of west. • How far west have you traveled in 2.5 minutes? • How far south have you traveled in 2.5 minutes?

Sample Problem • A roller coaster rolls down a 20o incline with an acceleration of 5.0 m/s2. • How far horizontally has the coaster traveled in 10 seconds? • How far vertically has the coaster traveled in 10 seconds?

Sample Problem • A particle passes through the origin with a speed of 6.2 m/s in the positive y direction. If the particle accelerates in the negative x direction at 4.4 m/s2 • What are the x and y positions at 5.0 seconds? • What are the x and y components of velocity at this time?

Vectors & 2-Dimensional Motion