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P2Week4 - P2 Week 4 Motion in One Dimension with constant...

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P2 Week 4 Motion in One Dimension with constant acceleration Graphical methods Differentiation, slopes, maxima and minima Integration, area under curve
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Summary for constant acceleration linear motion x-x 0 t-t 0 v v 0 a Equation v = v o + a × (t - t 0 ) x - x 0 = 1 2 v + v o ( ) (t - t 0 ) x x o = v o (t - t 0 ) + 1 2 a (t - t 0 ) 2 v 2 = v 0 2 + 2 a x x o ( )
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y-y 0 t-t 0 v v 0 a Equation Summary for freely falling objects Up is positive Hence acceleration due to gravity = – g g is the magnitude of acceleration due to gravity Average sea-level value for g =9.8m/s 2 . v = v o g × (t - t 0 ) y y o = v o (t - t 0 ) 1 2 g (t - t 0 ) 2 v 2 = v 0 2 2 g y y o ( ) y - y 0 = 1 2 v + v o ( ) (t - t 0 )
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Clicker problem Raindrops fall 2000 m from a cloud to the ground. If they were not slowed by air resistance, how fast would the drops be moving when they struck the ground? Take g=10m/s 2 . A: 10 m/s B: 30 m/s C: 200 m/s D: 300 m/s E: None of the above
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Clicker problem Raindrops fall 2000 m from a cloud to the ground. If they were not slowed by air resistance, how fast would the drops be moving when they struck the ground? Take g=10m/s 2 . Use v 2 =v 0 2 +2 × 10 × 2000 m 2 /s 2 . Set v 0 =0. So v=200 m/s = 200 × 3600 s / 1000 m × km/hr = 720 km/hr Having drops fall on your head at this speed would be quite dangerous! Large raindrops hit terminal speeds of about 40 km/hr in the air.
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Clicker problem How far does the runner whose velocity-time graph is shown below travel in 16 s? A: 50 m B: 70 m C: 90 m D: 100 m E: 120 m
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(2s, 8m) (10s, 72m) (12s, 84m) (16s, 100m) Clicker problem How far does the runner travel in 16s?
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P2 week 4 Differentiation, slopes, maxima and minima; Integration, area under curve
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Derivatives of elementary functions
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Clicker question Evaluate A. cos(x)-sin(x) B. cos(x)+sin(x) C. 2 cos(x) D. 2 sin(x) E. 0 d dx (sin( x ) + cos( x ))
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Clicker question Variables x and t are independent. Evaluate A. cos(t) B. sin(t) C. sin(x) D. 0 E. Indeterminate d dt sin( x )
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Clicker question Evaluate A. cos(x 2 ) B. sin(x 2 ) C. 2 x cos(x 2 ) D. x 2 cos(x 2 ) E. x 2 sin(x 2 ) d dx sin( x 2 )
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Chain rule If f is changing with x as df/dx and x is changing with t as dx/dt, then f is changing with t as This chain rule can be extended to include more “links” such as df dx dx dt df dx dx dy dy dt
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Chain rule example Find the derivative of f=cos(x 2 +wx) with respect to x. Let u=x 2 +wx Then f=cos(u) df dx = df du du dx = sin( u ) × (2 x + w ) = sin( x 2 + wx ) × (2 x + w )
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Chain rule example Find the derivative of f=cos 2 (x 2 +wx) with respect to x. Let u=x 2 +wx and v=cos(u) Then f=v 2 df dx = df dv dv du du dx = (2 v ) × ( sin( u )) × (2 x + w ) = 2 × cos( x 2 + wx ) × sin( x 2 + wx ) × (2 x + w )
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By parts d dy A ( y ) B ( y ) ( ) = A ( y ) d dy B ( y ) + B ( y ) d dy A ( y ) dA ( y ) dy d dy A ( y ) and are the same things Note: d dy y 2 sin( y ) ( ) = y 2 d dy sin( y ) + sin( y ) d dy y 2 = y 2 cos( y ) + 2 y sin( y ) Example:
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Clicker question Evaluate A.
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