# In order to describe motion you need to describe the following properties.

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1 Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1-D path speeding up and slowing down In order to describe motion you need to describe the following properties. 1. Coordinate system relative to some origin. Ex1: x y coordinate system. By convention positive x is taken to be to the right and positive y is taken to be up. 2. Position (,, ) location relative to the origin. The arrow over a symbol indicates the variable is a vector quantity. This means that it has a direction associated with it as well as a magnitude. Quantities with magnitude but no direction are called scalars. Ex1: positive and negative positions in one and two dimensions. NOTE: Calculations with vectors in one dimension is the same as for scalars. However in two dimensions vectors must be treated in a new way, see chapter Distance (d) Total length of travel measured is meters. 4. Displacement (,, Δ ) change in position units: meters. Note: that delta is always final minus initial. Ex2: Jack In the Box, Academy, Cotton Patch. 5. Average speed = total distance / total time or units: m/s The bar over a variable indicates average. It is also acceptable to subscript the variable with avg to indicate average as done below. 6. Average velocity = displacement/time units m/s Ex3: Find the average speed and velocity for trip from Academy to CP to JIB if it takes 0.2hr Ex4: Repeat with round trip ending back at Academy in 0.4hr. Ex5: Find the total distance traveled and the average speed for a trip where you drive for 30min at 80km/hr, then 12min at 100km/hr, then spend 15min receiving your ticket, and finally drive 45min at 40km/hr. 7. Graphical analysis of average velocity Note the average velocity is in the same form as the slope of a line. and

2 Position Position time The slope of a chord on a position vs. time plot is the average velocity over that time interval. In the diagram above the slope of the dashed chord gives the average velocity between 2 and 7 seconds. Since the slope is positive the average velocity is in the positive direction. What if we wanted to know the exact velocity at 4 seconds? The average velocity between 2 and 7 seconds gives an approximation of the velocity at 4 seconds. A better approximation could be found by decreasing the time interval over which the average velocity is taken time Mathematically this is called the limit calculations of which are the basis for calculus. 8. If we take the limit in which the time interval goes to zero, we get the instantaneous velocity at the time. mathematically this is equivalent to the slope of the tangent line

3 Position time 9. We can define the average acceleration in the same manner. units m/s 2 This is the slope of a chord on a velocity versus time plot. 10. The instantaneous acceleration can then be found by Ex6: ±a and ±v leading to objects speeding up or slowing down. 2.5 Motion with constant acceleration. Instead of describing the average properties of an object s motion it would be far more useful if we could describe the instantaneous properties without the use of calculus. We will show this can easily be done for the case of an object moving with a constant acceleration. We will also assume for simplicity sake in our derivation that all motion is along the x-axis, thus we do not need to worry about the vector nature of acceleration, velocity, and position. The results can be generalized into two dimensions as we will see in chapter 4. Assume a = constant that is However we are free to pick the initial time so take t i = 0. We will now write v i as v o (read as v not) indicating this is the velocity of the object at time zero. We will also drop the f subscript on the final time.

4 or We could also start with the definition of average velocity: and again let t i = 0 so x i = x o or solving for x f Since the acceleration (slope of the velocity vs time graph) is constant the velocity must be a linear function so the average velocity can be written as Thus the position of an object as a function of velocity and time can be written as We said earlier to describe the motion of an object fully we must describe position, velocity, and acceleration. Notice our first equation does not involve position. Our second does not involve acceleration. It will be useful to combine these equations in such a way to derive an equation that does not include final velocity, and one without time. First to eliminate final velocity substitute into. Or To eliminate time we will start by solving for time and substitute it into Start with and solve for time

5 Now substitute into to give us Rearranging a bit leaves us Solving for the final velocity term gives In summary The key here is these equations only hold when the acceleration is constant! Ex7: A truck covers 40m in 8.5s while smoothly slowing down to a final speed of 2.8m/s. Find its original speed and its acceleration. Ex8: A drag racer starts at rest and accelerates at 10m/s 2 for 400m. Find the time it takes to travel the 400m and the final speed of the car. Ex9: A ball is thrown straight up with an initial speed of 7m/s. How high will it rise? Ex10: A ball is thrown upwards from the ground with an initial speed of 25m/s at the same time a ball is dropped from a building 15m tall. After how long will the balls be at the same height? Demo activity: Have students work in pairs. Have students calculate their reaction time by dropping a meter stick from some level and measure how far it falls before they catch it between their fingers. Wikipedia gives average reaction time to be approximately 0.19s.

6 Chapter 3 Vectors Recall we defined two types of quantities in physics: 1. Scalars which have magnitude only. In other words they are numerical quantities units but not direction. Examples include: mass, time, energy, distance. Scalars may be positive, negative or zero. 2. Vectors which have magnitude and direction. Indicated by an arrow over the symbol, or bold print in the text. 3. There is a subset of vectors called unit vectors. These are vectors that have a magnitude of one and are used to indicate direction.. Example: Define a position vector to be. The magnitude of the vector is =3m and the direction is North East or = 45 above the x axis. In cases in which it is understood that you are describing the magnitude of a vector the absolute values and the vector can be left off,. Graphically we represent vectors using arrows, where the length of the arrow represents the magnitude of the vector. + y axis θ +x axis All vectors can be broken in to a pair of perpendicular scalar components using basic trigonometry.

7 y θ x Here and are the scalar components of. Notice that,, and form a right triangle of scalar quantities. Thus the Pythagorean Theorem relates the magnitude of a vector to its components by. Then recalling the definitions of Sine, Cosine and Tangent,,, and We see in this case the components can be found by: since it is always the magnitude of the vector. So, Similarly you can show NOTE: The x component does not always go with cosine, and the y with sine Ex1: Find the components of at 17 E of N Ex2: Find the components of at -100 Ex3: Given and calculate the magnitude and direction of

8 Ex4: Given and calculate the magnitude and direction of Note here that your calculator calculates angles ±90 from the ±x axis if you calculate the angle as.

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