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2. Mechanics Kinematics and Forces

3. Important things to remember 1) Units – Every numerical quantity must have units associated with it!! • Significant Figures – Important to show how precisely you know a measured quantity or a quantity calculated from measurements. • You should be using the correct number of significant figures at all times. • Problem Solving Steps: • Draw a Picture. • Choose your reference frame. • Identify known quantities. • Identify unknown quantities. • Identify equations that can be used to model this specific situation. • Solve the selected equation(s) for an unknown quantity. • Check numerical answer – for calculation errors. • Check units – use dimensional analysis. • Check significance of numerical value – Does it make sense? Make sure you understand the problems you are solving. It is not enough to determine a numerical value. Try to describe your reasoning in words. Ask yourself “why?” at each step.

4. CH 1: Vectors

5. Vector – quantity that has size and directional information. Magnitude – size of quantity, described by the length of the vector. Direction – Orientation of vector, described by an arrow. l q is often used as a numerical description of direction. Vector Notation = V q Magnitude of a vector A vector is drawn to represent a quantity within a specific coordinate system. There are three common coordinate systems, but these are by no means the only possible coordinate systems. Common Coordinate Systems This is the coordinate system we will typically use. Rectangular – Cartesian – x, y, z Cylindrical – r, q, z Spherical – r, q, f In 2D you have plane polar, which uses r and q. The choice of which coordinate system you will be using is up to you. You should choose a coordinate system such that it simplifies the solution to the problem. You can always transform a vector from one coordinate system to another.

6. Transforming from one coordinate system to another y How can we relate r or q to x and y? x (x, y) Pythagorean Theorem r y y q Converts from Cartesian to polar coordinates. x (0, 0) x r is standard notation for a length that can contain x, y and z coordinates. You must know how to use these trigonometric functions!! Remember that positive angles are measured counterclockwise from 0o (usually from the positive x- axis). These definitions are based on the angle shown in the diagram. If you use a different angle you may have to modify these expressions.

7. Vector Properties - Only when both magnitude and direction are exactly the same! - These must be added vectorally! Draw first vector. Draw second vector from tip of first vector (repeat for each additional vector present) Draw resultant (R) from tail of first vector to tip of last vector. - Cumulative law of addition. - Associative law of addition. - Negative of a vector. Same magnitude but opposite direction. - Subtraction of a vector. Same rules as for addition. • Multiply a vector by a scalar. • Positive scalar changes magnitude. • Negative scalar changes magnitude and direction.

8. Vector Components Vectors can be broken up into components. Components are used to describe part of the vector along each of the coordinate directions for your chosen coordinate system. - The number of components is determined by the number of dimensions. A 2D vector in the x-y plane has 2 components - 1 in the x direction and 1 in the y direction. A 3D vector in spherical coordinates has 3 components - 1 in the r direction, 1 in the q direction and 1 in the f direction. - All vector components are defined to be orthogonal (perpendicular) to each other. - Vector components are also vector quantities. Vectors can be moved as long as their length and orientation do not change! y These are only valid for the specified angle. When you square a vector, the result is a square of the magnitude. Directional information is removed. q x

9. Vector notation using unit vectors A unit vector is a dimensionless vector with a magnitude of 1 that is used to describe direction. Cartesian coordinate unit vectors: The following notation is also used: x – direction = y – direction = z – direction = x – direction = y – direction = z – direction = - The hat ‘^’ is used to distinguish unit vectors from other vectors. Examples of vectors using unit vector notation: Ax = magnitude of A along x – direction Ay = magnitude of A along y – direction Az = magnitude of A along z – direction x = length along x – direction y = length along y – direction z = length along z – direction

10. Example: Adding vectors using vector notation You add together magnitudes along similar directions. Remember to include appropriate sign. Rx Rz Ry You can determine magnitude and direction for 3D vectors in a similar fashion to what was done for 2D. qx, qy and qz are the angles measured between the vector and the specified coordinate direction. The magnitude can be found using the Pythagorean Theorem.

11. Examples: 1) A car is traveling 40 m/s at 20o west of south. What are the horizontal and vertical components of the car’s velocity. Write your answer in vector form. y x v vy q vx 2) A person walks 10 m at 15o N of E, then walks 30 m at 35o W of N. Determine the displacement of the person. y q2 D D2 q D1 q1 x from –x – axis or

13. Dot Product (Scalar Product) This product of two vectors results in a scalar quantity. You multiply one vector by the component of the second vector that is parallel to the first vector. If A = B: We use the same rules when multiplying a vector by itself. The square of a vector only gives magnitude.

14. Example: Determine the angle between the vectors A and B.

15. Vector Product (Cross-product) There are two different methods for determining the vector product between any two vectors: The Determinant method and the Cyclic method Determinant Method Cyclic Method Rewrite the j term as to get an identical expression

16. The direction of the resultant vector, for a right-handed coordinate system, for a vector product can be determined using the right-hand rule. Fingers point in the direction of the second vector The vector product looks at the product of two vectors which are perpendicular to each other, and who are also perpendicular to the resultant vector. Thumb points in the direction of first vector. Palm points in the direction of resultant vector. Example: Determine the magnitude and direction of the area of a parallelogram described by the vectors r1 and r2 which are used to describe the length of the two sides. y r1 r2 x Out of the page

17. CH 2: 1D motion

18. When we discuss motion what are we talking about? Translation – motion from one point to another along a straight line. } Rotation – motion along a curved path – circular motion Discussed Later Vibration – oscillatory motion Kinematics – The study of motion. This include all the types of motion listed above, but is most commonly used with translation. Translational motion can occur in one, two or three dimensions. We will begin with a discussion of one dimensional motion. What quantities do we use to describe motion? What do these quantities describe? Position - Location at a specific instant in time. Velocity - The rate at which the position of an object changes with time. - The rate at which the velocity of an object changes with time. Acceleration - An arbitrary quantity originally associated with the motion of the sun and moon. Time All motion can be described in terms of these quantities. Remember all the equations that we will be using are mathematical models that describe data collected based on observations. Models can be changed as new evidence becomes available.

19. Position - Location relative to a defined reference point at any instant in time. If you know the position of an object at every instant in time you know everything about the motion of the object. What two quantities are used to describe one position relative to another? Distance - the length of the path traveled between two points. Displacement - the net distance covered from the initial point to the final point. Units: Dx – Displacement xf = x – Final position xi = x0 –Initial position [d] = [Dx] = m, km, ft, in, mi, etc. [t] = s The symbol x is commonly used to denote position – it represents a location along the x-axis. If you look at position along a different coordinate axis you will often change x to y or z. A positive x value represents a location a specified distance away from the origin along the positive x direction (in the direction of the arrow). y x A negative x value represents a location a specified distance away from the origin along the negative x direction (in the opposite direction of the arrow). -1 1 0 2D Coordinate axis

20. Example: Start 100 Yards x0 = 0 For each case shown what is the distance traveled and what is the displacement? d = Dx = 100 Yards d = Dx = 150 Yards 200 Yards d = Dx = 50 Yards 100 Yards 0 Yards Notice that the distance and displacement do not have to be the same! Distance is what we call a scalar quantity, it only has a magnitude. This numerical value provides all the information you need to know about a distance. d = 150 Yards Displacement is a vector quantity, it has magnitude and direction. The displacement requires a numerical value and a direction. Dx = 50 Yards to the right

21. An object goes from one point in space to another. After it arrives at its destination, its displacement is: 1. either greater than or equal to 2. always greater than 3. always equal to 4. either smaller than or equal to 5. always smaller than 6. either smaller or larger than the distance it traveled.

22. Velocity - The rate at which the position of an object changes with time in a particular direction. There are two ways we discuss velocity. - estimated velocity over a relatively large time interval. Average velocity Instantaneous velocity - velocity at a particular instant of time (or over infinitesimally small time interval). Average velocity displacement total distance traveled total time time interval Average speed Average velocity in the x-direction Velocity is a vector quantity – requires magnitude and direction Speed is a scalar quantity – only has magnitude The average speed and the average velocity do not have to be the same!! When you use average velocity it does not mean the velocity is constant!! The average velocity contains no information about any fluctuations that occur in the instantaneous velocity.

23. Instantaneous velocity The instantaneous velocity can be determined by differentiating the expression that describes the position as a function of time. The time interval over which we are looking becomes infinitesimally small Units: [s] = [v] = m/s, km/hr, mi/hr, cm/s, etc. When discussing instantaneous velocity speed is the magnitude of the instantaneous velocity. Typically we will be using instantaneous velocity not average velocity, so from now on whenever we are discussing instantaneous velocity we will just say velocity. Example:

24. Example: Start 100 Yards x0 = 0 What is the average speed of the football player in the situation above if it takes 5 minutes to go from start to finish? What is the average velocity of the football player in the situation above if it takes 5 minutes to go from start to finish? To the right t = 5 min To the right = 300 s What is the instantaneous velocity of the football player at the opposite goal line? The football player must stop moving forward and begin moving in the reverse direction. v = 0 ft/s

25. A person initially at point P in the illustration stays there a moment and then moves along the axis to Q and stays there a moment. She then runs quickly to R, stays there a moment, and then strolls slowly back to P. Which of the position vs. time graphs below correctly represents this motion?

26. A marathon runner runs at a steady 15 km/hr. When the runner is 7.5 km from the finish, a bird begins flying from the runner to the finish at 30 km/hr. When the bird reaches the finish line, it turns around and flies back to the runner, and then turns around again, repeating the back-and-forth trips until the runner reaches the finish line. How many kilometers does the bird travel? 1. 10 km 2. 15 km 3. 20 km 4. 30 km The bird is traveling twice as fast as the runner.

27. Acceleration - The rate at which the velocity of an object changes with time in a particular direction. We also discuss acceleration in two ways: - estimated acceleration over a relatively large time interval. Average acceleration Instantaneous acceleration - acceleration at a particular instant of time (or over infinitesimally small time interval). Average acceleration Would a negative acceleration correspond to an increase or a decrease in the speed of an object? change in velocity in the x-direction It depends on the direction of the velocity. A negative acceleration means that there is an acceleration directed in the negative of the coordinate direction. time interval Average acceleration in the x-direction Instantaneous acceleration We will primarily discuss constant acceleration cases, but this is the general definition of acceleration. Do not mix time units!! – km/shr Units: [a] = m/s2, km/hr2, mi/hr2, cm/s2, etc.

28. Example: Start 100 Yards x0 = 0 What is the average acceleration of the football player in the situation above if it takes 5 minutes to go from start, where he is at rest, to finish, where he is traveling at a velocity of 1.5 ft/s to the left? To the left Is there an acceleration at the far goal line where the player stops for an instant? Yes!! – There is an acceleration any time the magnitude or the direction of the velocity changes!

29. 1 Dimensional Motion with a Constant Acceleration If the acceleration is constant the average and instantaneous accelerations are the same. 0 0 This equation describes how an initial velocity will change due to an acceleration applied over a period of time. Remember to include the appropriate sign for direction. } This is only valid for constant acceleration!! This equation describes how the initial position changes due to an initial velocity and a constant acceleration. These two equations can be used to model any translational motion under constant acceleration.

30. It is often useful to have an expression that does not rely on time. We can obtain such an expression by combining the two previously derived expressions. This expression relates velocity, position and acceleration without considering time.

31. Example: The driver of a car traveling at 30 m/s suddenly realizes there is a cliff 200 m in front of him. The brakes of the car can support a maximum acceleration of 2 m/s2 to slow down. Will he stop in time? v a

33. Graphical Analysis of motion What do the following graphs show? The object initially at rest starts at an initial position of 50 m and does not move from that spot. Position changes quadratically The object starts at an initial position of 50 m and an initial velocity of 10 m/s. The velocity of the object does not change. Position changes linearly The object initially starts at a velocity of 10 m/s from an initial location of 50 m, and the velocity increases at a rate of 2 m/s2. Velocity changes linearly

34. Analysis of a position vs. time graph Equation of a line. What can be determined from a position vs. time graph? The slope of a line connecting the initial and final point will provide average velocity information. The slope of the a tangent line to the curve at a point will provide us with instantaneous velocity information. Note the difference between the two lines! Analysis of a velocity vs. time graph What can be determined from a velocity vs. time graph? The slope of line connecting the initial and final point will provide average velocity information. The slope of the a tangent line to the curve at a point will provide us with instantaneous velocity information. Note that the average and instantaneous velocities are the same. This is only because the acceleration is constant!!

35. A train car moves along a long straight track. The graph shows the position as a function of time for this train. The graph shows that the train: 1. speeds up all the time. 2. slows down all the time. 3. speeds up part of the time and slows down part of the time. 4. moves at a constant velocity. The slope of the curve (or slope of the tangent line to the curve) decreases in time.

36. The graph shows position as a function of time for two trains running on parallel tracks. Which is true? 1. At time tB, both trains have the same velocity. 2. Both trains speed up all the time. 3. Both trains have the same velocity at some time before tB. 4. Somewhere on the graph, both trains have the same acceleration. The slopes of the two curves (or the tangent lines to the curves) are parallel at some point prior to tB.

37. Freefall – a specific example of 1D motion What is freefall? Motion towards the Earth only due to gravitational attraction. Close to the surface of the Earth it is commonly discussed as vertical motion. Gravitational attraction causes an object to be accelerated towards the surface of the Earth. Near the surface of the Earth this acceleration is assumed to be constant. It is not actually constant it changes with altitude and latitude. • If we assume up is a positive direction the direction of g would be negative. • You can define down as a positive direction. • Make sure you specify your positive coordinate directions. • Once you choose your directions for a specific problem do not change them!!

38. A ball is thrown upward with an initial speed of 20 m/s. What is the velocity, and acceleration at point 0? What is its velocity and acceleration at point 1 after 1 s? What is its velocity and acceleration at point 2? What is its velocity and acceleration at point 3, 1 s after point 2? What is its velocity and acceleration at point 4? What is its velocity and acceleration at point 5, 3 s after point 2? v = 20 m/s and a = -9.8 m/s2 2 v = 10.2 m/s and a = -9.8 m/s2 3 1 v = 0 m/s and a = -9.8 m/s2 v = -9.8 m/s and a = -9.8 m/s2 4 0 v = -20 m/s and a = -9.8 m/s2 5 v = -29.4 m/s and a = -9.8 m/s2

39. If you drop an object in the absence of air resistance, it accelerates downward at 9.8 m/s2. If instead you throw it downward, its downward acceleration after release is 1. less than 9.8 m/s2. 2. 9.8 m/s2. 3. more than 9.8 m/s2.

40. A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit the ground below the cliff with the greater speed is the one initially thrown 1. upward. 2. downward. 3. neither—they both hit at the same speed.

41. You are throwing a ball straight up in the air. At the highest point, the ball’s 1. velocity and acceleration are zero. 2. velocity is nonzero but its acceleration is zero. 3. acceleration is nonzero, but its velocity is zero. 4. velocity and acceleration are both nonzero.

42. A cart on a roller-coaster rolls down the track shown below. As the cart rolls beyond the point shown, what happens to its speed and acceleration in the direction of motion? 1. Both decrease. 2. The speed decreases, but the acceleration increases. 3. Both remain constant. 4. The speed increases, but acceleration decreases. 5. Both increase. 6. Other

43. CH 3: Motion in 2D

44. Do more objects move in 1D or 2D? 2D – most object do not move in a completely straight line with no side to side motion or up and down motion. • Examples of 2D motion: • car going around a corner • moves forward and to the right at the same time • car traveling straight on a rough road • moves forward and up and down at the same time • thrown object • moves forward and up or down at the same time • merry – go – round • moves forward and to the right at the same time What do you notice that is similar about the examples listed above? All of them refer to “at the same time”. This is the means by which we link the two different motions together. 2D motion is really just two different 1D motions. These two distinct motions are often not related. The mathematical models we use to describe 2D motion are the same ones we used for 1D motion. The main difference would be notation.

45. Position, Velocity and Acceleration in 2D Position in 2D Displacement in 2D Velocity in 2D Acceleration in 2D There are no changes to the definitions of these quantities, but we must remember that there are two components for each vector!!

46. 2D Motion with Constant Acceleration Differentiating vectors The constant acceleration equations are derived in an identical fashion to the way they were derived in 1D. They will not be re-derived here, but rewritten to include 2D. The subscripts show the direction we are working with, so we can ignore the unit vectors while working with the equations in this form. or and and or The x and y directions are completely independent of each other!!

47. Example: A particle moves in a horizontal xy-plane. It’s x and y positions as a function of time are given below. Write an expression for the a) position, b) velocity and c) acceleration in vector form. a) b) c)

48. Projectile Motion – an application of 2D motion We will use 2D motion to model the trajectory (path) a projectile follows as it travels through the air. • We are going to use two assumptions to simplify the problem: • g is a constant over the range of motion. • This is accurate as long as the range of motion is small compared to the radius of the Earth. • Air resistance is negligible. • This is Not justified for any real world example, but it is close for low speeds. • We are considering an ideal case to learn the basics. The trajectory is modeled using horizontal and vertical position. Both the horizontal and vertical positions change in time. What do these assumptions tell us about the vertical and horizontal motion? Vertical motion – Constant acceleration – the vertical motion is parabolic in time. Horizontal motion – Zero acceleration – the horizontal motion is linear with time and the velocity is constant. Since the accelerations for each direction of motion are different we can look at the two different directions of motion separately. How do we link the two motions? Both vertical and horizontal motion occurs simultaneously, therefore we can link the two distinct motions with time.

49. x y t t V Vy y Vx What would this motion look like if you rotated the x-axis so it was directed towards you? What would this motion look like if you looked down on it from above? y y x The actual motion is a combination of both of these motions occurring simultaneously. We can treat the vertical and horizontal motions as separate 1D motion. x We use the same 1D constant acceleration motion equations to model 2D constant acceleration motion. x

50. It is also sometimes useful to model the trajectory of a projectile by finding an equation that provides the vertical position based on the horizontal position. We will assume a known initial velocity v0 and a known launch angle q. 0 0 0 Equation of a parabola This shows us that the trajectory of an object will be parabolic. Remember the initial assumptions we made are still being used.