What we need to know… • Add, subtract, and resolve displacement and velocity vectors, so we can: • determine components of a vector along two perpendicular axes • determine the displacement and location of a particle relative to another • determine the change in velocity of a particle or the velocity of one particle relative to another • Understand the motion of projectiles in a uniform gravitational field, so we can: • Determine the horizontal and vertical components of velocity and position as functions of time • Analyze the motion of a projectile that is projected with an arbitrary initial velocity
What is a vector? • Objects do not always move in a straight line • To be accurate in analysis, you need two things… the value measured and direction • A vectorsare physical quantities that have both a magnitude and a direction
What is a vector? • Magnitudeis the scalarpart of a vector Scalar – A physical quantity that has no direction… just a number and units • Distance and Speed are scalars • “3 km”, “30 m/s” • Displacement, velocity, and acceleration are vectors • “3km north”, “30 m/s at 60 degrees”, a = 5i + 4j + 3k
Symbology • In the book • Scalars are in italics…. v • Vectors are in boldface with an arrow over the variable…. • On the board and in your work • Vectors have an arrow over the variable symbol… • In PPT, bold or with an arrow
a Vectors • Graphically • Vectors are depicted as arrows • Length is relative to magnitude • Arrowhead indicates direction • Since Vectors depict a magnitude and direction • Vectors can be moved graphically as long as the magnitude and direction are unchanged
R=a+b b b Graphic Vector Addition • Triangle Method • Tip to Tail • Resultant • A vector representing the sum of two or more vectors a
b R=a+b b Graphic Vector Addition • Parallelogram Method • Tail to Tail • Vectors can be added in any order a+b = b+a a
Adding Vectors • Vectors add in any order
-B B A A-B -B Vector Subtraction • A – B = A + (-B) -B has equal magnitude but has the opposite direction • Does A – B = B – A?
B B B 3B Scalar Multiplication • AB • Resultant is a vector in the same direction • Magnitude is A times the magnitude of B • Example:3B=?
Try this… • An A-10 normally flying at 80 km/hr encounters wind at a right angle to its forward motion (a crosswind). Will the plane be flying faster or slower than 80 km/hr?
Resultant Add the vectors 80 km/hr 60 km/hr 80 km/hr Is the resultant greater than or less than 80 km/hr? 60 km/hr Crosswind
Y X 2-D Coordinate System • To analyze motion, we need a frame of reference (FOR) • In 2-D, a good reference is the x-y coordinate plane • In our FOR, the vector has a magnitude, r, and a direction angle, q r q Remember… we can move a vector as long as we don’t change magnitude or direction
Y X Analyzing vectors • Once you establish a coordinate system or frame of reference, you can begin to analyze vectors mathematically • First step… resolve the vector into its x-component and its y-component • The component vectors can represent the change in x and the change in y for the vector r q Dy Dx
Y X Resolving Components • If we know the magnitude, d, and direction, q, we can find the x and y components using Trigonometry r Dy q Dx
Y X Unit Vectors • Any vector can be represented as the vector sum of its components • “Unit vectors” are used to specify the direction for each component • Magnitude equals “1” • “i” represents the x direction, “j” represents y direction (“k” will represent the z direction) r q Dy Dx
Unit Vectors • Once the components for two or more vectors are determined: • To add the vectors, simply add like components of each vector A B C For motion when direction is changing… we can use 1-D motion to analyze the components …then add the components to get the resulting motion R
Y X Determining Magnitude • To find the magnitude, r, of the Resultant vector, we use the Pythagorean Theorem: r Dy Dx
Y X Determining Direction • We can use the tangent function to determine the value of q: r Dy q Dx If using a calculator, be sure to set degrees or radians as appropriate
Y X Try this… • How fast must a truck travel to stay directly beneath and airplane that is moving 105 km/hr at an angle of 25 degrees to the ground? V=105 kph q=25o Vtruck=?
Try this… • A ranger leaves his base camp for a ranger tower. He drives on a heading of 125o for 25.5 km and then drives at a heading of 65o for 41.0 km. What is the displacement from the base camp to the tower?
Summary • Analyzing Vectors… • Resolve vectors into components “i”, “j”, “k” • Analyze Components (addition, motion…) • Add components to get new Resultant • We will use these steps to help us analyze motion in 2-D…
Projectile Motion • The most common example of an object which is moving in two-dimensions is a projectile • A projectile is an object upon which the only force acting is gravity • Projectile Motion-motion observed by any object which once projected is influenced only by the downward force of gravity. (provided that the influence of air resistance is negligible)
Projectile Motion • There are a variety of examples of projectiles: • an object dropped from rest is a projectile • an object which is thrown vertically upwards is also a projectile • an object is which thrown upwards at an angle is also a projectile
y Velocity Vertical Component x Horizontal Component Projectile Motion • Here is a typical projectile… a baseball thrown in the air • Remember… Any vector can be resolved into two perpendicular component vectors
y Velocity Vertical Component x Horizontal Component g Projectile Motion • Once the ball leaves your hand … it is only influenced by the acceleration of gravity • This is the definition of projectile motion
Projectile Motion • Since gravity acts along the vertical axis, the horizontal component of velocity is unaffected by gravity
y Velocity Vertical Component x Horizontal Component g Resolving Components • If we know the magnitude and direction of the projectile’s velocity, you can find the x and y components: v q
y Velocity Vertical Component x Horizontal Component g Projectile Motion • In the horizontal direction, acceleration is zero and velocity is constant
y Velocity Vertical Component x Horizontal Component g Projectile Motion • In the vertical direction, acceleration is constant (g) • All the equations derived for 1-D motion apply.
y Velocity Vertical Component x Horizontal Component g Projectile Motion • Vertical and Horizontal velocities are independent • Timeis the factor that relates the two components • To solve projectile motion problems.. You must use the given info to find the time, t.
Try this… • You throw a ball horizontally at 20 m/s from a 10 m tower. How far will the ball go before it hits the ground?
Projectile Path • Since the vertical distance component is related to time squared (y=1/2gt2), projectiles will follow a parabolic path. (note t = x/vcos)
Demo… A projectile is shot with velocity, v, at an angle, q. Assuming no air resistance… • How far, x, will it travel from the launch site? • What is the maximum distance it will travel? • What angle will give it the maximum distance? • At what two angles will give it travel the same distance? • What is the maximum height at each angle?
Do… A cannon will shoot a projectile with a muzzle velocity of 320 m/s. • What is the cannon’s maximum range? (10450m) • What angles will give a range of 3000. m? (8.3o, 81.7o) • What is the max height reached at each angle? (109m, 5100m)
Projectile Path Without Gravity… a projectile would follow the dotted path With Gravity… the projectile falls beneath this line the same vertical distance that it would if dropped from rest
Satellites V < 8000 m/s V almost 8000 m/s V = 8000 m/s V > 8000 m/s A satellite is simply a projectile that is constantly falling toward earth 8000m/s at an altitude of about 190 km
Summary • Analyzing 2-D Motion… • Resolve displacement, velocity, or acceleration vectors into components • Use 1-D equations of motion to determine motion along each component • Add components to get Resultant motion • Projectile motion • Motions in x and y directions are independent • Time relates the motion along each axis and must be determined
What we know… • Add, subtract, and resolve displacement and velocity vectors, so we can: • determine components of a vector along two perpendicular axes • determine the displacement and location of a particle relative to another • determine the change in velocity of a particle or the velocity of one particle relative to another • Understand the motion of projectiles in a uniform gravitational field, so we can: • Determine the horizontal and vertical components of velocity and position as functions of time • Analyze the motion of a projectile that is projected with an arbitrary initial velocity