EGR 1980 Recitation Worksheet 5 Introduction to Engineering Mechanics H Griffith 3 March 2014 Introduction In class we have introduced several topics related to both quadratic and linear expressions Such expressions are especially useful in describing the motion of an object in space In general the field of physics concerned with describing the motion of an object is referred to as kinematics Today we ll introduce three fundamental kinematic quantities 1 Displacement 2 Velocity and 3 Acceleration In addition we will explore mathematical models which describe the kinematics of some basic systems including one dimensional motion under constant acceleration deceleration as well as well as projectile motion Introduction to Basic Kinematic Quantities In order to describe motion we must begin by defining a quantity which describes how far an object travels from its original position over a period of time When thinking about such a quantity you should recognize that a complete description requires both the distance that the object has traveled as well as the direction which the object traveled For example traveling approximately 60 miles on Interstate 70 can lead to either Richmond Indiana or Columbus OH depending upon the direction of travel Since two pieces of information magnitude and direction are required to completely describe the quantity it is described mathematically as a vector The displacement vector describes the net change in position of an object over an interval of time Graphically we may represent the displacement vector as an arrow whose length describes the amount or magnitude of the displacement and whose orientation describes the direction of the displacement In the SI system of units the magnitude of the displacement is measured in meters For example a 10 meter displacement to the east and a 20 meter displacement to the north may be described by the following vectors d1 d2 Note that vector quantities are represented by either bold faced variable designators or by using the following half arrow notation d 1 d1 You try Draw a vector representing a 25 m displacement to the northeast Your vector should maintain the scale of those presented above To truly work with the complexities of vector quantities we must wait until we master some skills from trigonometry Therefore for the time being we will focus on displacement which occurs only along 1 dimension An example of such motion is an object dropped from rest from a specific height Notice that in this case motion occurs only along the vertical direction and is thus one dimensional Another example of one dimensional motion is a body car individual etc moving in a straight line In the case where motion occurs in only 1 direction we may use algebraic signs ie positive or negative to completely describe the direction of the vector quantity By convention for horizontal motion positive displacement is described as that occurring to the right ie east while negative displacement is described as that occurring to the left ie west Note that an object may undergo the same displacement in many different ways For example the displacement may occur over many different time periods ie a 1 m east displacement could occur over a 1 minute or 1 hour time interval To distinguish these different displacements we may introduce a quantity which describes the rate at which the displacement is occurring In general the average rate of change of any quantity is simply the change in that quantity divided by the time interval over which the change occurred Thus for displacement in one dimension the average rate of change may be defined as follows average rate of change of displacement The average rate of change of the displacement is referred to as the average velocity From observation we may see that the SI units for average velocity are meters per second Average velocity is denoted by the variable v with a superscript line ie v x t x t You should realize that just like displacement velocity is also a vector quantity However for the case of 1 dimensional motion we may represent its direction by simply using signed numbers Sign convention for velocity is consistent with that described above for displacement Also it is important to note that the speed of an object is defined specifically as the magnitude of the velocity and is therefore a scalar quantity You try A car undergoes a displacement of 100 meters to the east over a 5 second interval Calculate the average velocity of the car over this interval of time and draw your result as a vector Also calculate the speed of the object over this time interval How is your answer different from a car which travels to the west for the same distance and time interval In a similar fashion we may define the average rate of change of the velocity of an object in motion This quantity is referred to as the acceleration of the objection and is defined as shown below a v t Note that the units of acceleration are meters per second per second which is more simply written as meters per second squared m s2 Like displacement and velocity acceleration is also a vector quantity For motion in one dimensions direction is represented using a sign convention identical to that described above for displacement Motion in One Dimension Under Constant Acceleration Based upon your own physical intuition you should realize that the average quantities described above over a certain time interval are not necessarily equal to the instantaneous value of the quantity at any point in time during the interval For example consider your own travel on a long trip While you maintain an average speed of 50 mph there are certainly points in time which you were traveling either greater or less than 50 mph For example the velocity of an object may vary as follows over an interval of time v m s t s You should note that in the above graph the sign of the velocity remains positive over the entire time interval Therefore at any point in time the velocity vector still points to the right recall that for one dimensional motion the sign is used to indicate the direction of the vector indicating that the object is underdoing a displacement to the right over the entire time interval ie the direction of motion does not change However the magnitude of the velocity vector continuously decreases over the period of motion Therefore according to the above definition of speed we may say that the object is slowing down This type of kinematics could be used to