Describing motion: Kinematics in two dimension Scientist Galileo Galilei Issac Newton Vocabulary Vector scalars Resultant Displacement Components Resolving vectors Unit vector into its components Average Velocity Instantaneous Average acceleration Vector Projectile motion period velocity vector Centripetal acceleration vector Radial acceleration Parallelogram method Displacement Vector Instantaneous acceleration vector Frequency 3-1 Vectors and Scalars Quantities like velocity have direction as well as magnitude and thus is a vector quantity Other vector quantities are displacement, force, and momentum Mass, time, temperature have no direction associated with them- these are scalars When drawing vectors the arrow is always drawn so that it points in the direction of the vector quantity it represents The length of the arrow is drawn proportional to the magnitude of the vector quantity When we write the symbol for the vector we will always use boldface type If one is only concerned with the magnitude of the vector then we simply write the v in italics 3-2 Addition of Vectors- Graphical Method The resultant displacement is the sum of the vectors (that s why direction is important) You always lay the vectors head to tail Know the direction of the vector and keep it in that direction Parallelogram method- tail to tip method with more than two vectors 3-3 Subtraction of Vectors and Multiplication of a vector by a scalar This occurs when your vectors are in the negative direction Basically you do the same method- the difference in the vectors will be between the positive vector and the negative vector (just direction) 1 RoessBoss 2012-08-15 12:18:27 1/12 Kinematics 2D.pdf (#3)
A vector can be multiplied by a scalar c (basically you are increasing or decreasing the vector by a factor that is non directional 2 RoessBoss 2012-08-15 12:18:27 2/12 Kinematics 2D.pdf (2/12)
3-4 Adding Vectors by Components You can take a vector and deconstruct it into its components (x and y) using the concepts that support Pythagorean Theorem when they are at a 90 degree. Also we can use our trig functions to discover the components Trig Functions for finding the Components for angles other than 90 Equation Box 3-1 Note that there are two ways to specify a vector in a given coordinate system 1. We can give its components Vx and Vy 2. We can give its magnitude V and the angle theta it makes with the positive x axis We can also use Pythagorean Theorem Equation Box 3-2 And we can also use the definition of Tangent Equation Box 3-3 We will not add x components to y components. So when we have a component that has both an x and y (aka it is not fully horizontal or vertical) then we need to get the x out for each part and add together for the overall V in the x direction (also for the y) 3-5 Unit Vectors A unit Vector has a magnitude exactly equal to one It is useful to define unit vectors that point along the coordinate axes They are useful when adding vectors analytically by components 3 RoessBoss 2012-08-15 12:18:27 3/12 Kinematics 2D.pdf (3/12)
3-6 Vector Kinematics Displacement is the change in position The displacement vector is defined as the vector representing change in position (delta r) Equation Box 3-4 Average velocity vector can be redefined with the delta r Equation Box 3-5 Instantaneous velocity vector can also use r as a method to describe displacement better Equation Box 3-6 Average acceleration vector Equation Box 3-7 Instantaneous acceleration vector 4 RoessBoss 2012-08-15 12:18:27 4/12 Kinematics 2D.pdf (4/12)
Equation Box 3-8 5 RoessBoss 2012-08-15 12:18:27 5/12 Kinematics 2D.pdf (5/12)
Constant Acceleration- Chapter 2- now for 2 Dimensions X component- horizontal Y component- vertical To write the first two equations more formally you would have Equation Box 3-9 Equation Box 3-10 3-7 Projectile Motion 2Dimension motion like footballs, baseballs etc- thrown Projectile motion- take place in two dimensions, x and y Most cases we will ignore air resistance Our first case that we will examine is Galileo s case of an object thrown off of a table versus dropped from a table Remember that g= 9.80 m/s2 Galileo showed that by analyzing the horizontal and the vertical components of motion you could understand the motion that the object undertook. Horizontal and Vertical motion is analyzed separately in the projectile motion 6 RoessBoss 6/12 Kinematics 2D.pdf (6/12)
Vertical motion acceleration in the y direction is always constant- we generally set that to be a=g unless otherwise noted Horizontal motion the acceleration is generally 0, this is due to the velocity in the x direction being constant so v=constant in the x direction Galileo predicted that an object projected horizontally will reach the ground in the same time as an object dropped vertically. You can see that this is true with a multiple exposure photograph or you can conduct experiments in the lab to verify this If an object it projected upward at an angle the analysis is similar to the previous except that now there is an initial vertical component of velocity Because of the downward acceleration of gravity, vy continually decreases until the object reached the highest point on the path (so vyo= some number) When the object is at the highest point vy= 0 Then the vy starts to increase in the downward direction (that is it is becoming more negative) Vx remains constant The constant acceleration is that of gravity alone and is acting downward For an object projected at an upward angle the acceleration is one which (Vy) is continuously changing whereas the other Vx stays constant. When we look at an object projected at an upward angle, the acceleration is one (constant) direction, whereas the velocity has two components, one of which (Vy) is continuously changing, whereas the other Vx stays constant So Ay= -g = -9.80 m/s Note also that if the angle is known then the initial velocity has two components Equation Box 3-11 Now rewrite the equations for horizontal and vertical motion Equation box 3-12 Kinematic Equations for projectile motion (y positive upward; Ax= 0, Ay= -9.80 m/s^2) Horizontal Motion Vertical Motion Ax=0, Vx= constant Ay= -g = constant 7 RoessBoss 7/12 Kinematics 2D.pdf (7/12)
If y is taken positive downward, the minus sign becomes positive 3-8 problem solving with projectiles 1. Always read carefully and draw a careful diagram 2. Choose an origin and an xy coordinate system 3. Analyze the horizontal (x) motion and the vertical (y) motion separately. 4. If you are given the initial velocity you may want to resolve it into its x and y components 5. List the knowns and the unkowns quantities choosing Ax=o and Ay = -g or +g where g=9.80 m/s2, depending on whether you choose y positive up or down. Remember that Vx never changes throughout the trajectory, and that Vy=0 at the highest point to any trajectory, and that Vy=0 at the highest point of any trajectory thst returns downward. The velocity just before hitting the ground is generally not zero. 6. Think for a moment before jumping into the equations. A little planning goes a long way. Apply the relevant equations, combining equations if necessary. You may need to combine components of a vector to get magnitude and directions. Side note: Projectile motion is actually a parabola 3-9 Uniform circular motion An object that moves in a circle at constant speed v is said to undergo uniform circular motion A ball at the end of a rope Nearly uniform circular motion of the Moon around the earth. The magnitude of the velocity remains constant in the case but the direction of the velocity is continually changing Acceleration is defined as the rate of change of velocity, a change in direction of velocity means an acceleration is occurring, just as does a change in magnitude An object undergoing uniform circular motion is accelerating even if the speed remains constant (V1=V2). So when we look at it V is the change in velocity during the short time interval t. We will eventually consider what happens as it approaches zero and thus obtains instantaneous acceleration 8 RoessBoss 8/12 Kinematics 2D.pdf (8/12)
Equation Box 3-12 As it moves in its path during the time t, the small distance is l along the arc which subtends a small angle theta Since t is small so are l and the angle This acceleration is called centripetal acceleration- center seeking acceleration- or radial acceleration (since it is directed along the radius towards the center of the circle) and we denote it by Ar Next determine the magnitude of the centripetal (radial) acceleration Essentially when you dissect the arc you end up with a triangle centered from the radius Since the angle is so small we can say V = V1 = V2 because the magnitude of the velocity is assumed not to change Equation Box 3-13 This is an exact equality when t approaches zero, for then the arc length l equals the cord length (pg 64 for the diagram) Since we want to find the instantaneous acceleration for which t approaches zero. We will write the Equation from Box 3-13 as an equality and solve for V Equation Box 3-14 To get centripetal acceleration Ar, we divide V by t 9 RoessBoss 9/12 Kinematics 2D.pdf (9/12)
Equation Box 3-15 And since Equation Box 3-16 Is the speed, V of the object, we get To summarize- an object moving in a circle of radius r with constant speed V has an acceleration whose direction is toward the center of the circle and whose magnitude is Equation Box 3-17 It is not surprising that the acceleration depends on V and r For the greater the speed V, the faster the velocity changes direction, and the larger the radius, the less rapidly the velocity changes direction The acceleration vector points toward the center of the circle The velocity vector always points in the direction of motion, which is tangential to the circle Thus acceleration and velocity are perpendicular to each other at every point in the path of uniform circular motion So see acceleration and velocity do not have to pointed in the same direction For an object falling A and V are parallel In circular motion A and V are not parallel- nor are they in projectile motion where a=g is always downward but the velocity vector can have various directions 10 RoessBoss 10/12 Kinematics 2D.pdf (10/12)
Circular motion is often defined in terms of the frequency or so many revolutions per second (or other given time frame) The Period of an object revolving in a circle is the time required for one complete revolution Period and frequency are related by Equation Box 3-18 11 RoessBoss 11/12 Kinematics 2D.pdf (11/12)
If we look at the relationship to Velocity we can write it as (the circumference is 2πr) Equation Box 3-19 3-10 Relative Velocity When velocities are along the same line, simple addition or subtraction is sufficient to obtain the relative velocity But if they are not along the same lines we must make vector additions. It is important to know your frame of reference so that you can place them in the right order. You can use plus and minus signs to denote the direction of the vector. 12 RoessBoss 12/12 Kinematics 2D.pdf (12/12)