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Motion Chap along a straight line
Note this is an outline of the lecture, print this and take notes on it in lecture It is not the lecture. complete lecture. You should attend lecture to get the complete notes Images taken from Halliday/Resnick/Walker, Fundamentals of Physics, 7th Edition, John Wiley and Sons Inc., by permission.
Hints on HW
19 (c) When you differentiate x(t) to find its maximum value you will end up with a quadratic equation. This equation has two solutions. Choose the correct one! 31. Read off x0 directly from the graph. Now apply x x0 = v0t + at2 to two other points (you should get two equations that you can solve for a and v0). 61. Integrate v = dx/dt to find distance
Position and Displacement
Displacement (1-d) = final position - initial position
Average velocity (1-d)
x = x2 - x1
vavg =
x x2 - x1 = t t2 - t1
Average speed
savg =
total distance t
Example
Position of armadillo as a function of time
Example
For t = 1 4 s
vavg = x 6 m = = 2 m/s t 3s
savg =
total distance t 6m = = 2 m/s 3s
They are the same
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SP2-1 Drive along a straight road for 8.4 km at 70 km/h, then you run out of gas and stop. Takes 30 min to walk 2.0 km farther along the road to a gasoline station. (a) What is your overall displacement from the beginning of your drive to your arrival at the station? (b) What is the time interval t from the beginning of your drive to your arrival at the station? (c) What is your average velocity vavg from the beginning of your drive to your arrival at the station? Find it both numerically and graphically. (d) Suppose that to pump the gasoline, pay for it, and walk back to the truck takes you another 45 min. What is your average speed from the beginning of your drive to your return to the truck with the gasoline?
Discussion
Can vavg > 0 when savg = 0 ?
Can vavg = 0 when savg > 0 ?
You walk around the circumference of a circle, radius r = 2 m in 8 seconds. vavg = ?, savg = ?
Instantaneous velocity
Slope of x vs t at particular point:
SP2-2
See text
x dx v = lim = t dt t 0
Acceleration
Average acceleration
SP2-4
aavg =
v2 - v1 v = t2 - t1 t
A particle's position on the x axis of Figure 2-1 is given by
x = 4 - 27t + t 3
Instantaneous acceleration
v dv a = lim = t 0 t dt
Go back to SP2-2 and determine a(t)
with x in meters and t in seconds. (a) Find the particle's velocity function v(t) and acceleration function a(t). (b) Is there ever a time when v = 0? (c) Describe particle's the motion for t 0.
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Constant acceleration(a)
If a = constant Then average and instantaneous a are the same
Constant acceleration-equations v - v0 a = aavg = t -0 v = v0 + at
x - x0 t -0 x = x0 + vavg t vavg = vavg = v + v0 2
Constant acceleration-continued
Use : v = v0 + at , vavg = v + v0 x - x0 and vavg = 2 t -0 to get x = x0 + v0t + 1 at 2 (does not contain v) 2
Constant acceleration summary
Equations for Motion with Constant Accelerationa
Use : x = x0 + v0t +
at 2 and v = v0 + at 2 (does not contain t )
to get v 2 = v02 + 2a( x - x0 )
SP2-5
Spotting a police car, you brake a Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration. ( ) (a) What is that acceleration? (b) How much time is required for the given decrease in speed?
Another look at accel = constant dv a = v = adt = a dt = at + C dt
At t = 0, let the velocity be v0 v0 = a(0) + C = C v = v0 + at Now use v = dx to get x = x0 + vot + 1 at 2 2 dt
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Free Fall
acceleration a points downwards
Air resistance-Demo
1) Drop a lead weight and a wood block (the lead is about 6 times heavier). They both hit the ground at the same time. 2) Drop a lead weight a a piece of paper. The lead weight hits ground first.
magnitude is g = 9.8 m/s2 it d i 98 / (if air resistance is not important)
3) Ai resistance is important if an object has a low Air i i i bj h l mass and a large surface area. 4) 1968 Olympics Mexico city Bob Beamon (USA), 8.90 m. Record stood for 23 years. Why? After this his next best jump was 8.33 m.
SP2-6
September 26, 1993: Dave Munday went over Niagara Falls in a steel ball, fell 48 m to the water (and rocks). Assume his initial velocity was zero, neglect effect air (a) How long did Munday fall to reach the water surface? (b) Determine his position at each full second. (c) What was his velocity as he reached the water? (d) What was Munday's velocity at each count of one full second? Was he aware of his increasing speed?
SP 2-7
A pitcher tosses a baseball up along a y axis, with an initial speed of 12 m/s. (a) How long does the ball take to reach its maximum height? (b) What is the ball's maximum height above its release point? (c) How long does the ball take to reach a point 5.0 m above its release point?
Graphs and motion
a= dv dt
t1 t0
v1 - vo = adt = area under an a versus t graph v= dx dt
t1 t0
x1 - xo = vdt = area under a v versus t graph
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