Chapter 2 Kinematics: Description of Motion


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1 Chapter 2 Kinematics: Description of Motion Scalars A scalar quantity is a quantity that has magnitude only and has no direction in space Examples of Scalar Quantities: } Length } Area } Volume } Time } Mass 1 Vectors A vector quantity is a quantity that has both magnitude and a direction in space Examples of Vector Quantities: } Displacement } Velocity } Acceleration } Force 2.1 Distance and Speed: Scalar Quantities Distance is the path length traveled from one location to another. It will vary depending on the path. Distance is a scalar quantity it is described only by a magnitude Distance and Speed: Scalar Quantities Average speed is the distance traveled divided by the elapsed time: Distance and Speed: Scalar Quantities Since distance is a scalar, speed is also a scalar (as is time). Instantaneous speed is the speed measured over a very short time span. Question 2.1 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, have you and your dog traveled the same distance? a) yes b) no 6 1
2 and Velocity: Vector Quantities A vector has both magnitude and direction. Manipulating vectors means defining a coordinate system, as shown in the diagrams to the left. Displacement is a vector that points from the initial position to the final position of an object. 7 8 Note that an object s position coordinate may be negative, while its velocity may be positive; the two are independent. and Velocity: Vector Quantities For motion in a straight line with no reversals, the average speed and the average velocity are the same. Otherwise, they are not; indeed, the average velocity of a round trip is zero, as the total displacement is zero! 9 10 Different ways of visualizing uniform velocity: This object s velocity is not uniform. Does it ever change direction, or is it just slowing down and speeding up?
3 Question 2.2 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the a) yes park, do you and your dog have the same b) no displacement? Question 2.2 Displacement Does the displacement of an object a) yes depend on the specific location of b) no the origin of the coordinate system? c) it depends on the coordinate system Acceleration Question 2.2 Velocity in One Dimension If the average velocity is nonzero over a) yes some time interval, does this mean that b) no the instantaneous velocity is never zero c) it depends during the same interval? Acceleration is the rate at which velocity changes Acceleration Acceleration means that the speed of an object is changing, or its direction is, or both. 2.3 Acceleration Acceleration may result in an object either speeding up or slowing down (or simply changing its direction)
4 2.3 Acceleration If the acceleration is constant, we can find the velocity as a function of time: Question 2.3 Position and Speed a) yes If the position of a car is b) no zero, does its speed have to c) it depends on be zero? the position Kinematic Equations (Constant Acceleration) From previous sections: Question 2.4 Cruising Along I You drive for 30 minutes at 30 mi/ hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip? a) more than 40 mi/hr b) equal to 40 mi/hr c) less than 40 mi/hr Kinematic Equations (Constant Acceleration) Substitution gives: 2.4 Kinematic Equations (Constant Acceleration) These are all the equations we have derived for constant acceleration. The correct equation for a problem should be selected considering the information given and the desired result. and:
5 Vector Diagrams Vector Diagrams Vector diagrams are diagrams which use vector arrows to depict the direction and relative magnitude of a vector quantity. Vector diagrams can be used to describe the velocity of a moving object during its motion Describing Motion with Position vs. Time Graphs The Meaning of Shape for a pt Graph 27 To begin, consider a car moving with a constant, rightward (+) velocity  say of +10 m/s. Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a positiontime graph. Constant Velocity 28 Changing Velocity The position vs. time graphs for the two types of motion  constant velocity and changing velocity (acceleration)  are depicted as follows. Now consider a car moving with a rightward (+), changing velocity (acceleration)  that is, a car that is moving rightward but speeding up or accelerating Constant Velocity Positive Velocity Positive Velocity Changing Velocity (acceleration)
6 Importance of slope If the velocity is constant, then the slope is constant (i.e., a straight line). If the velocity is changing, then the slope is changing (i.e., a curved line). If the velocity is positive, then the slope is positive (i.e., moving upwards and to the right). Slope of p vs t Slow, Rightward (+) Fast, Rightward (+) Constant Velocity Constant Velocity Slope Slow, Leftward () Fast, Leftward () Constant Velocity Constant Velocity Meaning of slope Negative () Velocity Leftward () Slow to Fast Fast to Slow Determining the Slope on a pt Graph The slope of the line is +10 meter/1 second. It is obvious that in this case the slope of the line (10 m/s) is the same as the velocity of the car In this part of the lesson, we will examine how the actual slope value of any straight line on a graph is the velocity of the object. Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The next diagram depicts such a motion
7 Now consider a car moving at a constant velocity of +5 m/s for 5 seconds, abruptly stopping, and then remaining at rest (v = 0 m/s) for 5 seconds. Determining the slope The line is sloping upwards to the right. But mathematically, by how much does it slope upwards per 1 second along the horizontal (time) axis? To answer this question we must use the slope equation Check your understanding Answer: 3.0 m/s The Meaning of Shape for a vt Graph Consider a car moving with a constant, rightward (+) velocity  say of +10 m/s. As learned in an earlier lesson, a car moving with a constant velocity is a car with zero acceleration Note that a motion described as a constant, positive velocity results in a line of zero slope (a horizontal line has zero slope) when plotted as a velocitytime graph. Furthermore, only positive velocity values are plotted, corresponding to a motion with positive velocity. Now consider a car moving with a rightward (+), changing velocity  that is, a car that is moving rightward but speeding up or accelerating
8 The velocity vs. time graphs for the two types of motion  constant velocity and changing velocity (acceleration)  can be summarized as follows Positive Velocity Zero Acceleration Positive Velocity Positive Acceleration Notice that the slope of a velocitytime graph represents the acceleration of the object Now how can one tell if the object is speeding up or slowing down? Speeding up means that the magnitude (the value) of the velocity is getting large Question 2.5 You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? Throwing Rocks I a) the separation increases as they fall b) the separation stays constant at 4 m c) the separation decreases as they fall d) it is impossible to answer without more information Question 2.5 Throwing Rocks II You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities? a) both increase at the same rate b) the velocity of the first rock increases faster than the velocity of the second c) the velocity of the second rock increases faster than the velocity of the first d) both velocities stay constant 47 8
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