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Kinematic Graphs. Graphically exploring derivatives and integrals. Slope. Y. (4,8). 8. (0,0). X. 4. Slope = rise = Y = Y2 – Y1 = 8 – 0 = 8 = 2 run X X2 – X1 4 – 0 4. Velocity is the slope of Displacement. Y. 8. (4,8). Displacement
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Kinematic Graphs Graphically exploring derivatives and integrals Dr. Sasho MacKenzie - HK 376
Slope Y (4,8) 8 (0,0) X 4 Slope = rise = Y = Y2 – Y1 = 8 – 0 = 8 = 2 run X X2 – X1 4 – 0 4 Dr. Sasho MacKenzie - HK 376
Velocity is the slope of Displacement Y 8 (4,8) Displacement (m) (0,0) X 4 Time (s) Average Velocity = rise = D = D2 – D1 = 8 – 0 = 8 m = 2 m/s run t t2 – t1 4 – 0 4 s Dr. Sasho MacKenzie - HK 376
The displacement graph on the previous slide was a straight line, therefore the slope was 2 at every instant. • Which means the velocity at any instant is equal to the average velocity. • However if the graph was not straight the instantaneous velocity could not be determined from the average velocity Dr. Sasho MacKenzie - HK 376
Instantaneous Velocity • The average velocity over an infinitely small time period. • Determined using Calculus • The derivative of displacement • The slope of the displacement curve Dr. Sasho MacKenzie - HK 376
Instantaneous Acceleration • The average acceleration over an infinitely small time period. • Determined using Calculus • The derivative of velocity • The slope of the velocity curve Dr. Sasho MacKenzie - HK 376
Y The average velocity does not accurately represent slope at this particular point. 8 (4,8) Displacement (m) (0,0) X 4 Time (s) Average vs. Instantaneous Average Velocity = rise = D = D2 – D1 = 8 – 0 = 8 m = 2 m/s run t t2 – t1 4 – 0 4 s Dr. Sasho MacKenzie - HK 376
Derivative • The slope of the graph at a single point. • Slope of the line tangent to the curve. • The limiting value of D/ t as t approaches zero. Dr. Sasho MacKenzie - HK 376
dt t3 t2 t1 Infinitely small time period (dt) • Tangent line • Instantaneous Velocity Displacement Time Dr. Sasho MacKenzie - HK 376
Displacement Velocity = 0 Velocity > 0 Velocity < 0 Time Velocity from Displacement The graph below shows the vertical displacement of a golf ball starting immediately after it bounces off the floor and ending when it lands again. Dr. Sasho MacKenzie - HK 376
Displacement Velocity 0 OR Velocity Acceleration 0 0 Graph Sketching Differentiation Slope + 0 Dr. Sasho MacKenzie - HK 376
Slope 0 _ + Displacement Velocity 0 0 0 OR Velocity Acceleration 0 Graph Sketching Differentiation Dr. Sasho MacKenzie - HK 376
Y (2,4) 4 (0,0) X 2 Going the other way: area under the curve Area under curve = Height x Base = Y x X = 4 x 2 = 8 Dr. Sasho MacKenzie - HK 376
Y (2,4) 4 Velocity (m/s) (0,0) Time (s) X 2 Displacement is the Area Under the Velocity Curve Displacement = V x t = 4 x 2 = 8 m Dr. Sasho MacKenzie - HK 376
(2,4) 4 This would be an over estimate of the area under the curve Velocity (0,0) 2 Time What if velocity isn’t a straight line? Dr. Sasho MacKenzie - HK 376
Integration • Finding the area under a curve. • Uses infinitely small time periods. • All the areas under the infinitely small time periods are then summed together. • D = Vdt = Vt Dr. Sasho MacKenzie - HK 376
The area under the graph in these infinitely small time periods are summed together. Velocity Time Instantaneous Velocity Infinitely small time period Infinitely small time periods • D = Vdt = V1t1 + V2t2 + V3t3 + …… Dr. Sasho MacKenzie - HK 376
Constant slope of 2 Displacement Velocity 0 OR 2 Velocity Acceleration t t t 0 Area under curve increases by the same amount for each successive time period (linear increase). Graph Sketching Integration Dr. Sasho MacKenzie - HK 376
Exponential curve Displacement Velocity 0 OR Velocity Acceleration t t t 0 Area under curve increases by a greater amount for each successive time period (exponential increase). Graph Sketching Integration Dr. Sasho MacKenzie - HK 376
Indicate on the acceleration graph below the location of the following point(s). Place the letter on the graph. A. Zero velocity B. Zero acceleration C. Max velocity D. Min velocity E. Max acceleration F. Min acceleration 2 0 -1 1 s 2 s 3 s Dr. Sasho MacKenzie - HK 376
Indicate on the velocity graph below the location of the following point(s). Place the letter on the graph. A. Zero velocity B. Zero acceleration C. Max velocity D. Min velocity E. Max acceleration F. Min acceleration G. Max displacement H. Min displacement 0 Dr. Sasho MacKenzie - HK 376