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Uniform Acceleration

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Uniform Acceleration. Motion in One Dimension. Sects. 5-7: Outline. 1. Kinematics of one dimensional motion with uniform acceleration:. 2. Special case: Freely falling objects  a = -g = - 9.8 m/s 2. 3. Special case: a = 0  motion with uniform velocity v = constant. Uniform Acceleration. - PowerPoint PPT Presentation

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Page 1: Uniform Acceleration

Uniform Acceleration

Page 2: Uniform Acceleration

1. Kinematics of one dimensional motion with uniform acceleration:

f iv v at

2. Special case: Freely falling objects a = -g = - 9.8 m/s2

1 22f i ix x v t at

2 2 2f iv v a x f ix x vt

3. Special case: a = 0 motion with uniform velocity v = constant

Motion in One DimensionSects. 5-7: Outline

Page 3: Uniform Acceleration

Uniform Acceleration • The derivation is in the text twice, once using algebra & once

using calculus. Read on your own! • Notation (drop subscripts x on v & a, for motion along x only!)

t 0 = time when the problem beginsxi initial position (at t = 0, often xi = 0)

vi initial velocity (at t1 = 0)

t time when we wish to know other quantitiesxf position at time tvf velocity at time t

a acceleration = constant

(Average & instantaneous accelerations are equal)

Page 4: Uniform Acceleration

Uniform Acceleration Equations • Results (one dimensional motion only!):

vf = vi + at (1)xf = xi + vi t + (½)a t2 (2)(vf)2 = (vi)2 + 2a(x - x0) (3)vavg = (½)(vf + vi) (4)

NOT VALID UNLESS a = CONSTANT!!!Often xi = 0. Sometimes vi = 0

Page 5: Uniform Acceleration

Physics and Equations • IMPORTANT!!!

– Even though these equations & their applications are important, Physics is not a collection of formulas to memorize & blindly apply!

– Physics is a set of PHYSICAL PRINCIPLES.– Blindly searching for the “equation which will

work for this problem” is DANGEROUS!!!!– On exams, you get to have an 3´´ 5´´ index card

with anything written on it (both sides) you wish. On quizzes, I will give you relevant formulas.

Page 6: Uniform Acceleration
Page 7: Uniform Acceleration

Example: Acceleration of Car

Known: x0 = 0, x = 30 m, v0 = 0, a = 2.0 m/s2

Wanted: t Use: x = (½)a t2

t = (2x/a)½ = 5.48 s

Page 8: Uniform Acceleration

Example: Estimate Breaking Distances

v = v0 = constant = 14 m/st = 0.50 s

a = 0x = v0t = 7 m

a = - 6.0 m/s2

v decreases from 14 m/s to zero

x0 = 7 m, v0 = 14 m/s, v = 0v2 = (v0)2 + 2a(x – x0)

x = x0 + [v2 - (v0)2]/(2a)= 7 m + 16 m = 23 m

Page 9: Uniform Acceleration

Example: Fastball

Known: x0 = 0, x = 3.5 m, v0 = 0, v = 44 m/s Wanted: a Use: v2 = (v0)2 + 2a (x - x0)

a = (½)[v2 - (v0)2]/(x - x0) = 280 m/s2 !

Page 10: Uniform Acceleration

Problem: A jet lands on an aircraft carrier at 140 mi/h (63 m/s).     a) Calculate the acceleration (assumed constant) if it stops in t = 2.0 s due to the arresting cable that snags the airplane & stops it.

b) If the plane touches down at position xi = 0, calculate it’s final position.

Example 2.7: Carrier Landing, p. 35

Page 11: Uniform Acceleration

Example 2.8, Watch Out for the Speed Limit! p. 39 Problem: A car traveling at a constant velocity of magnitude 41.4 m/s passes a trooper hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets out from the billboard to catch it, accelerating at a constant rate of 3.90 m/s2. How long does it take her to overtake the speeding car?

Page 12: Uniform Acceleration

Freely Falling Objects

Page 13: Uniform Acceleration

Freely Falling Objects • Important & common special case of uniformly

accelerated motion:

“FREE FALL”

Objects falling in Earth’s gravity. Neglect air resistance.

Use one dimensional uniform acceleration equations (with some changes in notation, as we will see)

Page 14: Uniform Acceleration

A COMMON MISCONCEPTION!

Page 15: Uniform Acceleration

• Experiment finds that the acceleration of falling objects (neglecting air resistance) is always (approximately) the same, no matter how light or heavy the object.

• Acceleration due to gravity, a gg = 9.8 m/s2 (approximately!)

Page 16: Uniform Acceleration

• Acceleration of falling objects is always the same, no matter how light or heavy.

• Acceleration due to gravity, g = 9.8 m/s2

• First proven by Galileo Galilei

Legend: Dropped objects off of the leaning tower of Pisa.

Page 17: Uniform Acceleration

• Acceleration due to gravityg = 9.8 m/s2 (approximately)

– Depends on location on Earth, latitude, & altitude:

Page 18: Uniform Acceleration

• Note: My treatment is slightly different than the book’s, but it is, of course, equivalent!

• To treat motion of falling objects, use the same equations we already have, but change notation slightly:

Replace a by g = 9.8 m/s2

– But in the equations it could have a + or a - sign in front of it! Discuss this next!

• Usually, we consider vertical motion to be in the y direction, so replace xf by yf and xi by yi (often y0i = 0)

Page 19: Uniform Acceleration

NOTE!!! Whenever I write the symbol g, it ALWAYS means the POSITIVE numerical value 9.8 m/s2! It NEVER is negative!!! The sign (+ or -) of the gravitational acceleration is taken into account in the equations we now discuss!

Page 20: Uniform Acceleration

Sign of g in 1d Equations • Magnitude (size) of g = 9.8 m/s2 (POSITIVE!)

– But, acceleration is a vector (1 dimensional), with 2 possible directions.

– Call these + and -. – However, which way is + and which way is - is

ARBITRARY & UP TO US!– May seem “natural” for “up” to be + y and “down” to be

- y, but we could also choose (we sometimes will!) “down” to be + y and “up” to be - y

– So, in equations g could have a + or a - sign in front of it, depending on our choice!

Page 21: Uniform Acceleration

Directions of Velocity & Acceleration • Objects in free fall ALWAYS have downward

acceleration.• Still use the same equations for objects thrown

upward with some initial velocity vi

• An object goes up until it stops at some point & then it falls back down. Acceleration is always g in the downward direction. For the first half of flight, the velocity is UPWARD.

For the first part of the flight, velocity & acceleration are in opposite directions!

Page 22: Uniform Acceleration

VELOCITY & ACCELERATION

ARE NOT NECESSARILY IN

THE SAME DIRECTION!

Page 23: Uniform Acceleration

Equations for Bodies in Free Fall • Written taking “up” as + y!

vf = vi - g t (1)yf = yi + vit – (½)gt2 (2)(vf)2 = (vi)2 - 2g(yf - yi) (3)vavg = (½)(vf + vi) (4)g = 9.8 m/s2

Often, yi = 0. Sometimes vi = 0

Page 24: Uniform Acceleration

Equations for Bodies in Free Fall • Written taking “down” as + y!

vf = vi + g t (1)yf = yi + vit + (½)gt2 (2)(vf)2 = (vi)2 + 2g(yf - yi) (3)vavg = (½)(vf + vi) (4)g = 9.8 m/s2

Often, yi = 0. Sometimes vi = 0

Page 25: Uniform Acceleration

Example

Note: y is positive DOWNWARD!

v = aty = (½) at2

a = g = 9.8 m/s2

v1 = (9.8)(1) = 9.8 m/s

v2 = (9.8)(2) = 19.6 m/s

v3 = (9.8)(3) = 29.4 m/s

Page 26: Uniform Acceleration

Example

Page 27: Uniform Acceleration

Problem: A stone is thrown at point (A) from the top of a building with an initial velocity of vi = 19.2 m/s straight upward. The building is H = 49.8 m high, and the stone just misses the edge of the roof on its way down, as in the figure. Answer these questions:

a) Calculate the time at which the stone reaches its maximum height.

b) Calculate the maximum height of the stone above the rooftop.

c) Calculate the time at which the stone returns to the level of the thrower

d) Calculate the velocity of the stone at this instant.

e) Calculate the velocity & position of the stone at time t = 5 s.

Example 2.12: Not a bad throw for a rookie! p. 38

Page 28: Uniform Acceleration

1. Kinematics of one dimensional motion with uniform acceleration:

f iv v at

2. Special case: Freely falling objects a = -g = - 9.8 m/s2

1 22f i ix x v t at

2 2 2f iv v a x f ix x vt

3. Special case: a = 0 motion with uniform velocity v = constant

Motion in One DimensionSects. 5-7: Summary