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2-2 Average Velocity

2-2 Average Velocity. Speed: how far an object travels in a given time interval. (2-1). Velocity includes directional information:. 2-3 Instantaneous Velocity. The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short. (2-3).

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2-2 Average Velocity

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  1. 2-2 Average Velocity Speed: how far an object travels in a given time interval (2-1) Velocity includes directional information:

  2. 2-3 Instantaneous Velocity The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short. (2-3) These graphs show (a) constant velocity and (b) varying velocity.

  3. 2-4 Acceleration Acceleration is the rate of change of velocity.

  4. 2-4 Acceleration There is a difference between negative acceleration and deceleration: Negative acceleration is acceleration in the negative direction as defined by the coordinate system. Deceleration occurs when the acceleration is opposite in direction to the velocity.

  5. 2-4 Acceleration The instantaneous acceleration is the average acceleration, in the limit as the time interval becomes infinitesimally short. (2-5)

  6. 2-5 Motion at Constant Acceleration We can also combine these equations so as to eliminate t: We now have all the equations we need to solve constant-acceleration problems. (2-10) (2-11a) (2-11b) (2-11c) (2-11d)

  7. Kinematics is the description of how objects move with respect to a defined reference frame. • Displacement is the change in position of an object. • Average speed is the distance traveled divided by the time it took; average velocity is the displacement divided by the time. • Instantaneous velocity is the limit as the time becomes infinitesimally short.

  8. Average acceleration is the change in velocity divided by the time. • Instantaneous acceleration is the limit as the time interval becomes infinitesimally small. • The equations of motion for constant acceleration are given in the text; there are four, each one of which requires a different set of quantities.

  9. 2-7 Falling Objects Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity. This is one of the most common examples of motion with constant acceleration.

  10. 2-7 Falling Objects In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance.

  11. 2-7 Falling Objects The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s2.

  12. 2-8 Graphical Analysis of Linear Motion This is a graph of x vs. t for an object moving with constant velocity. The velocity is the slope of the x-t curve.

  13. 2-8 Graphical Analysis of Linear Motion On the left we have a graph of velocity vs. time for an object with varying velocity; on the right we have the resulting x vs. t curve. The instantaneous velocity is tangent to the curve at each point.

  14. 2-8 Graphical Analysis of Linear Motion The displacement, x, is the area beneath the v vs. t curve.

  15. x x (m) (m) t t (s) (s) v v (m/s) (m/s) t t (s) (s) distance vs time graph constant velocity position changes over time at a constant rate constant position no change in position zero velocity velocity vs time graph constant acceleration velocity changes over time at a constant rate constant velocity no change in velocity zero acceleration

  16. 10 velocity/time graph 8 v (m/s) constant acceleration 6 constant deceleration constant velocity 4 II III constant velocity 2 I IV 0 50 20 40 10 30 t (sec)

  17. 10 velocity/time graph 8 I. Velocity is constant at 4 m/s Acceleration is 0 Time = 20 seconds Distance = 80 meters x = 80 m x=vt 4(20)=80 m v (m/s) constant acceleration 6 constant deceleration constant velocity 4 II III constant velocity 2 I IV 0 50 10 20 30 40 III. t (sec) Acceleration is constant deceleration at 0.8 m/s2 Velocity is changing Time = 10 seconds Distance = 60 meters Vf = 2 m/s Vi = 10 m/s t = t4 - t3 t = 40 - 30 = 10 sec a = (Vf - Vi)/t a = (2 - 10)/10 = -0.8 m/s2 (deceleration) x = vt x = (10+2) (10) = 60m 2 II. Acceleration is constant at 0.6 m/s2 Velocity is changing Time = 10 seconds Distance = 70 meters Vf = 10 m/s Vi = 4 m/s t = t3 - t2 t = 30 - 20 = 10 sec a = (Vf - Vi)/t a = (10 - 4)/10 = 0.6 m/s2 x = vt x = (10+4) (10) = 70m 2 IV. Velocity is constant at 2 m/s Acceleration is 0 Time = 10 seconds Distance = 20 meters t = t5 - t4 t = 50 - 40 = 10 sec x = 20 m x=vt 2(10)=20 m Total Distance 230 meters (80+70+60+20) Total Time 50 seconds Average Velocity 4.6 m/s (230/50)

  18. velocity/time graph Total Distance 230 m Total Time 50 seconds Average Velocity 4.6 m/s 10 8 II v (m/s) constant acceleration III 6 constant I deceleration constant velocity IV 4 V0 = 4 m/s Vf = 10 m/s a = 0.6 m/s2 t = 10 seconds x = 70 m V0 = 4 m/s Vf = 4 m/s a = 0 t = 20 seconds x = 80 m constant velocity V0 = 10 m/s Vf = 2 m/s a = 0.8 m/s2 t = 10 seconds x = 60 m 2 V0 = Vf = 2 m/s a = 0 t = 10 seconds x = 20 m 0 50 20 40 10 30 t (sec)

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