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Understanding Equations of Motion: Constant Velocity and Acceleration

This announcement discusses the principles of motion, focusing on the equations governing motion under constant velocity and constant acceleration. It highlights how to relate velocity to the slope of a position-time graph and how to determine position from a velocity-time graph. Key formulas for calculating changes in velocity and position, as well as techniques for visualizing acceleration on graphs, are provided. A practical example involving a car approaching an intersection illustrates the application of these concepts in decision-making scenarios.

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Understanding Equations of Motion: Constant Velocity and Acceleration

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  1. Announcements • No class nextMonday (MLK day)

  2. Equations of Motion Tractable cases §2.5–2.6

  3. Find Position from Velocity • Generally: velocity is slope of a position-time graph. • Conversely, position is the area under a velocity-time graph. • What is this when v is constant?

  4. distance units Area under a v-t graph area = (a m/s)(b s) = ab m speed (m/s) a b time (s)

  5. Constant-Velocity Motion • v = Dx/Dt = constant throughout process • Dx = vDt • xf = xi + Dx = xi + vDt • Can also use this with average v

  6. Find Velocity from Acceleration • General case: acceleration is slope of a velocity-time graph. • Conversely, velocity is the area under an acceleration-time graph. • What is this when a is constant?

  7. Constant-Acceleration Motion • Instantaneous accel = average accel • a = Dv/Dt • Dv = velocity change over time Dt • Dv = aDt • v = v0 + Dv = v0 + aDt

  8. Acceleration on an x-t Graph • Velocity is the slope of a position-time graph • Acceleration means a changing slope • A constant slope means a straight x-t line • A varying slope means a curved x-t line • Positive acceleration = concave up • Negative acceleration = concave down

  9. Visualize Acceleration Young and Freedman, Fig. 2.8 Board Work: • Signs of v • Signs of a

  10. slope = velocity d t slope = acceleration v area = distance t a area = velocity t Acceleration Starting from a traffic light that turns green

  11. Equations of Motion • What are velocity and position under conditions of constant acceleration?

  12. Formulas from Constant x-Acceleration • Velocity change Dv = aDt • Velocity vt = v0 + Dv = v0 + aDt • Position change Dx = v0Dt + 1/2 a (Dt)2 • Position xt = x0 + v0Dt + 1/2 a (Dt)2

  13. Another Form (constant a) • If you don’t know Dt and want v: x = x0 + v0Dt + 1/2a (Dt)2Dt = Dv/a x – x0 = v0 Dv/a + 1/2a (Dv/a)2 2a (x–x0) = 2v0 (v–v0) + (v–v0)2 2a (x–x0) = 2vv0 – 2v02 + v2 – 2vv0 + v02 2a (x–x0) = 2vv0 – 2vv0 + v2 + v02 – 2v02 2a (x–x0) = v2 – v02 v2 = v02 + 2a (x–x0) Do units work?

  14. Another Form (constant a) • If you don’t know a but know v, v0, and Dt: x = x0 + v0Dt + 1/2a (Dt)2a = Dv/Dt = (v–v0)/Dt x = x0 + v0 Dt + 1/2((v–v0)/Dt) (Dt)2 x – x0 = v0 Dt + 1/2v Dt – 1/2v0 Dt x – x0 = v0 Dt – 1/2v0 Dt + 1/2v Dt x – x0 = 1/2 (v0 + v)Dt Do units work?

  15. Example Problem A car 3.5 m in length traveling at 20 m/s approaches an intersection. The width of the intersection is 20 m. The light turns yellow when the front of the car is 50 m from the beginning of the intersection. If the driver steps on the brake, the car will slow at –3.8 m/s2 and if the car steps on the gas the car will accelerate at 2.3 m/s2. The light will be yellow for 3 s. To avoid being in the intersection when the light turns red, should the driver use the brake or the gas?

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