Advanced Physics Chapter 3 Kinematics in Two Dimensions; Vectors

Chapter Vectors and Scalars 3-2 Addition of Vectors—Graphical Methods 3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar 3-4 Adding Vectors by Components 3-5 Projectile Motion 3-6 Solving Problems Involving Projectile Motion 3-7 Projectile Motion is Parabolic 3-8 Relative Velocity

3-1 Vectors and Scalars Scalar A quantity that has only magnitude (size) and units Mass, time, distance, speed

3-1 Vectors and Scalars Vector A quantity that has both direction and magnitude. It also has units! Displacement, velocity, acceleration, force When drawing a vector quantity use an arrow whose length represents its size and points in the direction of the vector To represent a vector quantity in an equation, put an arrow over the letter symbol. Example: velocity = v

3-2 Addition of Vectors— Graphical Methods Two or more vectors that measure the same quantity can be added together to get their sum Resultant—sum of vectors

3-2 Addition of Vectors— Graphical Methods To add vectors graphically (The Butt-head Method): 1. Draw first vector to scale with in correct direction. 2. Draw the second vector with its butt (tail) to the head of the first vector. 3. Continue to draw additional vectors butt to head. 4. Connect the butt of the first vector with the butt of a resultant vector and the head of the last vector to the head of a resultant vector. 5. Measure the size and direction of the resultant vector; this is the sum of the vectors.

3-2 Addition of Vectors— Graphical Methods To add vectors graphically (The Parallelogram Method): 1. Draw first vector to scale with in correct direction. 2. Draw the second vector with its butt (tail) to the butt of the first vector. 3. Draw a parallelogram using these two sides. 4. Draw the resultant vector so that it is diagonal from the common origin. 5. Measure the size and direction of the resultant vector; this is the sum of the vectors.

3-2 Addition of Vectors— Graphical Methods To add two perpendicular vectors mathematically: 1. Draw first vector in correct direction. 2. Draw the second vector with its butt (tail) to the head of the first vector. 3. Connect the butt of the first vector with the butt of a resultant vector and the head of the last vector to the head of a resultant vector. 4. Find the size of the resultant using the Pythagorean Theorem. 5. Find the direction of resultant by using SOH, CAH, TOA trigonometry.

3-2 Addition of Vectors— Graphical Methods To add any two vectors mathematically: 1. Draw first vector in correct direction. 2. Draw the second vector with its butt (tail) to the head of the first vector. 3. Connect the butt of the first vector with the butt of a resultant vector and the head of the last vector to the head of a resultant vector. 4. Find the size of the resultant using the Law of Cosines c 2 = a 2 + b 2 – (2ab cos  ) 5. Find the direction of resultant by using the Law of Sines sin  /a = sin  /b = sin  /c or a/sin A = b/ sin B = c/ sin C Important!

3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar Subtraction of Vectors Use the butt-head method but reverse the direction of the second vector (or any subtracted) Multiplication of a Vector by a Scalar Increases the size of the vector by the magnitude of the scalar

3-4 Adding Vectors by Components A single vector can be thought of as being made up of two perpendicular vectors. These perpendicular vectors are called the vector’s components. Usually the components are chosen to be along the x and y axes. Finding the components of a single vector is called resolving the vector into its components

3-4 Adding Vectors by Components When adding vectors you can either us trigonometry or you may first find the x and y components of each vector, add the components together and then find the resultant of these two vectors. This is called adding vectors by components.

3-4 Adding Vectors by Components Sample problem: A mailperson leaves the post office and drives 22km due north and then 47km 60º south of east. Find her displacement from the post office. Use BOTH trigonometry and component addition

3-4 Adding Vectors by Components Sample problem: A mailperson leaves the post office and drives 22km due north and then 47km 60º south of east. Find her displacement from the post office. 30 km 38.5ºsouth of east

3-5 Projectile Motion Projectile Motion When an object moves through the air only under the force of gravity. Air resistance is ignored! The object has a horizontal and a vertical component of its motion. These horizontal and vertical components are independent of each other. When dealing with projectile motion each component is analyzed separately

3-5 Projectile Motion Horizontal Motion no net force acts on object so it doesn’t accelerate Initial velocity (horizontal) = final velocity (horizontal) Vertical motion A net force acts on the object so it accelerates it downward (g = 9.8 m/s 2 ) Velocity changes magnitude and/or direction as the object moves through the air

3-5 Projectile Motion Since all objects fall at same rate, a ball dropped or thrown horizontally from the same height will hit the ground at the same time!

3-5 Projectile Motion An object projected at an upward angle follows a path where: Horizontal component of velocity is constant Vertical component of velocity has same magnitude but different directions at two places. Vertical component of velocity is zero at top of path Acceleration due to gravity acts on vertical component only and it acts downward.

3-6 Solving Problems Involving Projectile Motion 1. Read problem and draw a diagram 2. Separate velocity vector into its horizontal and vertical components. 3. Write down all vertical and horizontal knowns and unknowns 4. Solve for vertical and horizontal components separately 5. Find the ½ time using vertical components 6. Use ½ time to find maximum height 1. (v top = 0, a = g = – 9.8 m/s 2) 7. Use ½ time to find hang time and then range

3-6 Solving Problems Involving Projectile Motion Sample Problem: A football is kicked at an angle of 37º with a velocity of 20.0 m/s. Find its: Hang time Maximum height Range

3-6 Solving Problems Involving Projectile Motion Sample Problem: A football is kicked at an angle of 37º with a velocity of 20.0 m/s. Find its: Hang time 2.45 s Maximum height 7.35 m Range 39.2 m

3-7 Projectile Motion is Parabolic Duh!

3-8 Relative Velocity If observations are made in different reference frames objects may appear differently to observers in each frame. Example #1 A person in a moving car drops a can out the window. To the person inside the car the can seems to fall backwards. To a person standing outside the car the can seems to fall forward.

3-8 Relative Velocity Example #2 A boat moves at a speed of 10 m/s straight across a river. The river flows perpendicularly to the boat a 5 m/s. How fast is the boat approaching a person floating in a lifejacket straight across from the boat. How fast is the boat moving relative to a person standing on the shore?

3-8 Relative Velocity Example #2 A boat moves at a speed of 10 m/s straight across a river. The river flows perpendicularly to the boat a 5 m/s. How fast is the boat approaching a person floating in a lifejacket straight across from the boat? 10m/s How fast is the boat moving relative to a person standing on the shore? 11.2 m/s