Kinematics Position, Velocity , and Acceleration Graphs

Overview Kinematics: A Description of Motion Position and Displacement Velocity Average Instantaneous Acceleration Graphing The topics we will discover on this powerpoint

A Story We will introduce Kinematics by looking at motion along a single line. This restricted perspective is usually called 1-Dimensional Motion. If we can answer the following Questions by using a both Algebra and Graphing, then we know we GROK 1-dimensioal kinematics: How far did you walk? What where your speeds at any location on your journey? What where your velocities at any location? What was your average Speed and Average Velocity? What was your Displacement? Tell me about your accelerations during the journey Imagine walking away from your house to visit an ice cream store that is located 2 km away. You walk at a constant speed and you arrive at the store in 30 minutes. You spend 10 minutes at the store eating your ice cream. You then leave the store and continue walking away from home at your normal pace for 5 minutes. You realise that you should be heading home and immediately walk back towards home at twice your normal pace. At a distance halfway between the ice cream store and your home you stop for 10 minutes at a library (good students always visit libraries). You then continued back home at your increased speed.

Position vs Time Plots Gives us the location, x, at any time, t. x 3 1 For example: Position at t=3, x(3) = 1 1 t 3 -3

Graph the Journey Get out some graph paper and Sketch a Position-Time graph of our story. Don’t forget to use those graphing skills you learned in Math class. Imagine walking away from your house to visit an ice cream store that is located 2 km away. You walk at a constant speed and you arrive at the store in 30 minutes. You spend 10 minutes at the store eating your ice cream. You then leave the store and continue walking away from home at your normal pace for 5 minutes. You realise that you should be heading home and immediately walk back towards home at twice your normal pace. At a distance halfway between the ice cream store and your home you stop for 5 minutes at a library (good students always visit libraries). You then continued back home at your increased speed.

You then leave the store and continue walking away from home at your normal pace for 5 minutes You then continued back home at your increased speed. Imagine walking away from your house to visit an ice cream store that is located 2 km away. You walk at a constant speed and you arrive at the store in 30 minutes You spend 10 minutes at the store eating your ice cream. You realise that you should be heading home and immediately walk back towards home at twice your normal pace. At a distance halfway between the ice cream store and your home you stop stop for 5 minutes at a library (good students always visit libraries). Compare your graph to mine. Let’s do some algebra to calculate distances and speeds at each line segment of the graph.

Kinematic Definitions Position (x) – where you are located Distance (d) – how far you have traveled, regardless of direction Displacement (x = xf-xi) – where you are in relation to where you started, need initial and final position. Displacement is a vector: It has both, magnitude and direction!! For one dimensional motion a +ve sign means the displacement is toward the right, a -ve sign means the displacement is toward the left. Distance is a scalar, only magnitude.

Kinematic Definitions Speed (v) – distance divided by time. It is always a positive quantity and the direction of motion is irrelevant. Speed is a scalar quantity. Velocity (v) – is displacement divided by time. Since displacement depends only on your starting and ending points, velocity (v) is x divided by t and thus a vector. If displacement is negative, then the velocity will also be negative. Similarly if displacement is positive, then the velocity will be positive.

Graphical Interpretation of Average Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions (A and D)

The average velocity you travelled at on your walk between home and the ice cream store is: We also notice that the total distance travelled by you is also 2km and the total amount of time it took you was also 30 min. So your average speed is also 0.0667 km/min 2km 30 min

The average velocity you travelled when sitting at the ice cream store eating your ice cream: We also notice that the total distance travelled by you is also 0 km and the total amount of time it took you was also 10 min. So your average speed is also 0 km/min 10 min

The average velocity you travelled for 5 minutes away from the ice cream store: 1/3 km 5 min We also notice that the total distance travelled by you is also 1/3 km and the total amount of time it took you was also 5 min. So your average speed is also 0.0677 km/min

The average velocity you travelled when heading back home: We also notice that the total distance travelled by you is also 1 1/3 km and the total amount of time it took you was also 10 min. So your average speed is also 0.1333 km/min 1 1/3 km 10 min

Note: Since we are returning home, look what happens to the average speed and average velocity from home to this location : We notice that the total distance travelled by you is 2 km +1/3 km + 1 1/3 km = 3.667 km and the total amount of time it took you was also 55 min. So your average speed is now 0.0667 km/min 1 km 55 min

The average velocity during your library: We also notice that the distance travelled by you is also 0 km and the total amount of time it took you was also 5 min. So your average speed is also 0 km/min 5 min

The average velocity final walk home: We also notice that the distance travelled by you is also 1 km and the total amount of time it took you was also 7.5 min. So your average speed is also 0.1333 km/min 1 km 7.5 min

The average velocity for the whole journey: We also notice that the distance travelled by you is now 4.667 km and the total amount of time it took you was also 67.5 min. So your average speed is 0.06913 km/min 67.5 min

Understanding An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is: either greater than or equal to always greater than always equal to either smaller or equal to either smaller or larger than the distance it traveled.

Example: Suppose that in both cases truck covers the distance in 10 seconds:

Instantaneous Velocity Instantaneous velocity is defined as the limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time This notation tells us that Instantaneous Velocity is a Derivative of position with respect to time. I invented the mathematics for this, and it is called Calculus. Why? Because I need two points to determine a slope, but I have only one point available:

Graphical Interpretation of Instantaneous Velocity Instantaneous velocity is the slope of the tangent to the curve at the time of interest The instantaneous speed is the magnitude of the instantaneous velocity The slope of the tangent line drawn at B, is the Instantaneous Velocity at B

Average vs Instantaneous Velocity Average velocity Instantaneous velocity

Understanding The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: at time tB both trains have the same velocity both trains speed up all the time both trains have the same velocity at some time before tB train A is longer than train B all of the above statements are true position A B Note: the slope of curve B is parallel to line A at some point t< tB tB time

Average Acceleration A velocity that changes indicates that an acceleration is present Average acceleration is the rate of change of the velocity Average acceleration is a vector quantity (i.e. described by both magnitude and direction) Note: Velocity also changes when the speed remains the same, but the direction changes. This is because velocity is a vector.

Average Acceleration When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are the opposite (either positive or negative), then the speed is decreasing Units SI Meters per second squared (m/s2) CGS Centimeters per second squared (cm/s2) US Customary Feet per second squared (ft/s2)

Instantaneous and Uniform Acceleration Instantaneous acceleration is the limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be uniform. That is, the instantaneous accelerations will all be equal to the average acceleration Yep, that Calculus thing again.

Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph. That is, the slope of secant line PQ Instantaneous acceleration is the slope of the tangent to the curve of the velocity - time graph, That is, slope of tangent line at t=b b

Velocity vs Time Plots Gives velocity at any time by just looking at the height of the graph at any time t. Net area gives displacement (how far you have moved from initial position to final position) Total area gives how far you have travelled Slope gives acceleration (rise over run). Speed s = | v | If the shapes you are calculating the area of is not simple, you can calculate the area using my Calculus. v (m/s) 9-1=8 3 9+1=10 t 4 -3

Let’s Draw a Velocity Time Graph based on out previous ice cream trip’s Position Time graph. Then we will use that to determine how far we walked. Slope (Velocity) is 0.06667 km/min Slope (Velocity) is -0.1333 km/min Slope (Velocity) is 0.06667 km/min Slope (Velocity) is 0 km/min Slope (Velocity) is 0 km/min Slope (Velocity) is -0.1333 km/min

Hey, you walked 2 km + 1/3 km= 2 1/3 km away from home and 1 1/3 km+ 1 km = 2 1/3 km back toward home. So our net distance travelled is 0km. While the total distance travelled was 2 1/3 km+ 2 1/3 km = 4 2/3 km Let’s Draw a Velocity Time Graph based on out previous ice cream trip’s Position Time graph. Then we will use that to determine how far we walked. 1/3 km 2 km 1 1/3 km 1 km

Acceleration vs Time Plots Gives acceleration at any time. Area gives change in velocity a (m/s2) The change in velocity between t=4 and t=1 is then 6+(-2)=4 m/s. 3 6 2 t 4 -3

Let’s Check your Understanding Is it possible for an object to have a positive velocity at the same time as it has a negative acceleration? 1 - Yes 2 - No If the velocity of some object is not zero, can its acceleration ever be zero ? 1 - Yes 2 - No What about negative acceleration and speeding up?

Velocity Understanding If the average velocity of a car during a trip along a straight road is positive, is it possible for the instantaneous velocity at some time during the trip to be negative? A - Yes B - No Drive north 5 miles, put car in reverse and drive south 2 miles. Average velocity is positive while the instantaneous velocity is negative when the car is going backwards. .

Graphical Relationships

Relating Graphs

Understanding A ball is dropped from a height of two meters above the ground. y x Draw vy vs t -6 v t 0.5 9 A B C D E

Understanding Draw v vs t Draw x vs t Draw a vs t x A ball is dropped for a height of two meters above the ground. t v Draw v vs t Draw x vs t Draw a vs t t a t

Tossed Ball A ball is tossed from the ground up a height of two meters above the ground. And falls back down. y x Draw v vs t v -6 t 1 9 A B C D E

Tossed Ball Draw v vs t Draw x vs t Draw a vs t x A ball is tossed from the ground up a height of two meters above the ground. And falls back down. t v Draw v vs t Draw x vs t Draw a vs t t a t

Understanding A ball is thrown straight up in the air and returns to its initial position. During the time the ball is in the air, which of the following statements is true? A - Both average acceleration and average velocity are zero. B - Average acceleration is zero but average velocity is not zero. C - Average velocity is zero but average acceleration is not zero. D - Neither average acceleration nor average velocity are zero. = 0 Vave = DY/Dt = (Yf – Yi) / (tf – ti) aave is not 0 since Vf and Vi are not the same ! . aave = DV/Dt = (Vf – Vi) / (tf – ti)

Example Where is velocity zero? Where is velocity positive? Where is velocity negative? Where is speed largest? position vs. time Where is acceleration zero? Where is acceleration positive? velocity vs. time

Final Check x(t) t v(t) v(t) v(t) t t t 4s Dx Dt 5s Slope of x vs t gives v Area under v vs t gives x! Which plot best represents v(t) 4s Dx Dt t 5s v(t) v(t) v(t) t t t

Matching Graphs