x-t Graphs

d-t graphs E F D C B A Constant speed Speeding UP Constant Speed (faster!) Slowing Down At rest t (sec) d (m) B C E A D F 10 15 20 27 36 120 100 50 30

sAB = rise/run = (30-0m) / (10-0s) = 3 m/s t (sec) d (m) B C E A D F 10 15 20 27 36 120 100 50 30 SLOPE AVERAGE Speed on a d-t graph can be found by taking the ______________. sAB = rise/run = (30-0m) / (10-0s) = 3 m/s sCD = rise/run = (100-50m) / (20-15s) = 10 m/s

Remember, d-t graphs are only valid when you are moving FORWARD Remember, d-t graphs are only valid when you are moving FORWARD. They can’t account for backward motion. This is why we “upgrade” to an x-t graph.

x-t graphs (an object moving forward) Constant speed Speeding UP Constant Speed (faster!) Slowing Down At rest t (sec) x (m) B C E A D F 10 15 20 27 36 120 100 50 30

x-t graphs (moving forward and backward) t (sec) x (m) t1 t2 t3 x2 x1 x3 B C D A Constant speed (Constant + velocity, or constant velocity in the + direction) Slow down, speed up, slow down, speed up 2 moments where the object is “at rest” (for a moment)

On an x-t graph, the slope of the graph is the … AVERAGE VELOCITY On an x-t graph, the slope of the graph is the … AVERAGE VELOCITY. (NOT NECESSARILY THE SPEED!!!)

If an x-t graph shows motion in the negative direction, what does the negative slope show? x (m) Slope = -5/16 Since slope is average speed, s = 5/16 m/s 6 No sign b/c it’s a scalar average velocity is -5/16 m/s 1 t (sec) Needs a sign b/c it’s a vector 4 20

Average speed and average velocity always have the same “number-value” on an x-t graph, right? NO!!!  The values are only the same when the object doesn’t change direction. Let’s look at a few examples …

Understand the difference between velocity and speed on an x-t graph. x (m) 10 20 30 40 50 t (s) 30 20 10 Example: What is the average speed from t = 10 to t = 40 seconds? 17 m 10 m dist10-40 = 27 m Avg. Speed = dist/ Dt = 27m / 30 sec = 0.9 m/s

Understand the difference between velocity and speed on an x-t graph. x (m) 10 20 30 40 50 t (s) 30 20 10 Example: What is the average velocity from t = 10 to t = 40 seconds? Dx10-40 = + 7 m Avg. Velocity = slope = Dx/ Dt = + 7 / 30 sec = + 0.23 m/s

x (m) 10 20 30 40 50 t (s) 30 20 10 Example: What is the average velocity from t = 20 to t = 40 seconds? Avg. vel. = slope = rise/run = -7 m / 20 = -.35 m/s Since the objects displacement is in the NEGATIVE direction, so is its average velocity.

x (m) 10 20 30 40 50 t (s) 30 20 10 Example: What is the average SPEED from t = 20 to t = 40 seconds? Avg. Speed = distance / time = 15 m / 20 = .75 m/s 4m 11m CAUTION!!! YOU CAN’T USE THE SLOPE TO FIND AVG SPEED ON AN X-T GRAPH. WHY?? Since the object changed direction, distance and displacement are NOT the same!!!

Will avg. velocity EVER be greater than avg. speed? NO!!! Will avg. velocity EVER be equal to avg. speed? YES!!! When the path travelled was one-way, in a straight line.