Announcements. No class next Monday (MLK day). Equations of Motion. Tractable cases. §2.5–2.6. 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 ?.
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Equations of Motion
area = (a m/s)(b s) = ab m
Young and Freedman, Fig. 2.8
slope = velocity
slope = acceleration
area = distance
area = velocity
Starting from a traffic light that turns green
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?
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?
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?