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ACCELERATION ACCELERATION CH2 SEC 2

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Page 1: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

ACCELERATIONACCELERATIONCH2 SEC 2

Page 2: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

ACCELERATION MEASURES ACCELERATION MEASURES THE RATE OF CHANGE IN THE RATE OF CHANGE IN VELOCITYVELOCITY

AVERAGE ACCELERATION EQUATION

if

if

avg tt

vv

t

va

changefor required time

yin velocit changeonaccelerati average

Page 3: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

EXAMPLE PROBLEMEXAMPLE PROBLEM A SHUTTLE BUS SLOWS TO A STOP WITH AN AVG

ACCELERATION OF -1.8 m/s2. HOW LONG DOES IT TAKE THE BUS TO SLOW FROM 9.0 m/s to 0.0 m/s?

Given: vi= 90 m/s vf= 0 m/s

aavg= -1.8 m/s2

Unknown: ∆t=? Solve: Rearrange the avg. acceleration equation to solve for the time

interval 1st change this: to:

2rd solve:

Answer:

t

vaavg

avga

vt

smsmsmvvv if /0.9/0.9/0

2/8.1

/0.9

sm

smt

st 0.5

Page 4: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

ACCELERATION HAS ACCELERATION HAS DIRECTION AND MAGNITUDEDIRECTION AND MAGNITUDE

Imagine a train is moving to the right (East), so that the displacement and the velocity are positive.

Velocity increases in magnitude as the train picks up speed. (so, when ∆v is positive, the acceleration is positive)=A

The train with no stops will travel for awhile at a constant velocity; because the velocity is not changing, ∆v= 0m/s.=B (when velocity is constant, the acceleration is equal to

zero) If the train still traveling in a positive direction, slows

down, the velocity is positive, but the acceleration is negative.=C

Page 5: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

RECOGNIZING MOTION ON A RECOGNIZING MOTION ON A GRAPHGRAPH

A= speed or velocity increasing

B= constant velocity

C= speed or velocity decreasing

Page 6: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

VELOCITY AND VELOCITY AND ACCELERATION TABLEACCELERATION TABLE

vI a Motion

+ + speeding up

- - speeding up

+ - slowing down

- + slowing down

+ or - 0 constant velocity

0 + or - speeding up from rest

0 0 remaining at rest

Page 7: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

MOTION WITH CONSTANT MOTION WITH CONSTANT ACCELERATIONACCELERATION

A ball moving in a straight line with constant acceleration. (image was captured 10 times in 1 second, so the time interval between images equals 1/10 of a second)

As the ball’s velocity increases,

the ball travels a greater distance

during each time interval. Because the acceleration is constant,

the velocity increases by the same

amount during each time interval. So, the distance that the ball travels

in each time interval is equal to the

distance it traveled in the previous

time interval, plus a constant distance.

Page 8: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

The relationships between displacement, velocity, and constant acceleration are expressed by equations that apply to any object moving with constant acceleration.

Page 9: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Displacement depends on Displacement depends on acceleration, initial velocity, and acceleration, initial velocity, and

timetimeWe know that the average velocity equals displacement divided by the time interval.

For an object moving with constant acceleration, the avg velocity is equal to the average of the initial velocity and the final velocity.

To find an expression for the displacement in terms of the initial and final velocity, we can set the expressions for average velocity equal to each other.

This equation can be used to find the displacement of any object moving with constant acceleration.

t

xvavg

2fi

avg

vvv

2fi

avg

vvv

t

x

Page 10: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Displacement with Constant Displacement with Constant Uniform Acceleration Uniform Acceleration

EquationEquation

tvvxfi )(

2

1

Displacement= ½ (initial velocity + final velocity)(time intervial)

Page 11: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Example QuestionExample QuestionA racing car reaches a speed of 42 m/s. It then

begins a uniform negative acceleration, using its parachute and braking system, and comes to a rest 5.5 seconds later. Find how far the car moves while stopping.Given: vi= 42 m/s vf= 0 m/s ∆t= 5.5 sUnknown: ∆x=?Solve: Use the equation for displacementtvvx

fi )(

2

1

)5.5)(/0/42(2

1ssmsmx

)5.5)(/21( ssmx

mx 120

Page 12: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Final VelocityFinal VelocityDepends on initial velocity, acceleration, Depends on initial velocity, acceleration,

and timeand timeWhat if final velocity is not known, but we still

want to calculate the displacement? (we can if we known the initial velocity, the uniform acceleration, and the elapsed time)

By rearranging the equation for acceleration, we can find a value for finial velocity.

Adding the initial velocity to both sides of the equation, we get an equation for the final velocity.

t

vv

tt

vva if

if

if

ifvvta

fivvta

Page 13: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Velocity with Constant Velocity with Constant Uniform AccelerationUniform Acceleration

tavvif

Final velocity= initial velocity + (acceleration x time intervial)

You can use this equation to find the final velocity of an object moving with uniform acceleration after it has accelerated at a constant rate for any time interval, whether the time interval is a minute or half an hour.

Page 14: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

If you want to know the displacement of an object moving with uniform acceleration over some certain time interval, you can obtain another useful expression for displacement by substituting the expression for vf into the expression for ∆x.

This equation is useful not only for finding displacement of an object with uniform acceleration, but also for finding the displacement required for an object to reach a certain speed or to come to a stop.

tvvxfi )(

2

1

ttavvxii

)(2

1

])(2[2

1 2tatvxi

2)(2

1tatvx

i

Page 15: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Example questionExample question A plane started at rest at one end of a runway

undergoes a uniform acceleration of 4.8 m/s2 for 15 s before takeoff. What is its speed at takeoff? How long must the runway be for the plane to be able to take off?

Given: vi= 0 m/s a= 4.8 m/s ∆t= 15 s Unknown: vf=? ∆x=?

Solve: Use the equation for the velocity of a uniformly accelerated object.

Part 2:Use the equation for the displacement

tavvif

)15)(/8.4(/0 2 ssmsmv

f

smvf

/72

2)(2

1tatvx

i

22 )15)(/8.4(2

1)15)(/0( ssmssmx

mx 540

Page 16: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Time Can Also Be FoundTime Can Also Be FoundMethod involves rearranging one equation to

solve for ∆t and substituting that expression in another equation.

Start with the equation for displacement.

We can sub this expression into the equation for final velocity.

tvvxfi )(

2

1

)2

(fivv

xt

)2

(fi

if vv

xavv

tavvif

Page 17: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Final Velocity After Any Final Velocity After Any DisplacementDisplacement

When using this equation, you must take the square root of the right side of the equation to find the final velocity.

xavivf

222

Page 18: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

Example ProblemExample ProblemA person pushing a stroller starts from rest,

uniformly accelerating at a rate of 0.500 m/s2. What is the velocity of the stroller after it has traveled 4.75m?Given: vi= 0m/s a= 0.500 m/s2

∆x= 4.75 mUnknown: vf=?

Solve: Use final velocity equation.

Substitute the values into the equation: Remember to take the square root in the final step.

xavivf

222

)75.4)(/500.0(2)/0( 222

msmsmvf

222

/75.4 smvf

smsmvf

/18.2/75.4 222 smv

f/18.2

Page 19: ACCELERATION CH2 SEC 2. ACCELERATION MEASURES THE RATE OF CHANGE IN VELOCITY AVERAGE ACCELERATION EQUATION

EQUATIONS FOR UNIFORMLY EQUATIONS FOR UNIFORMLY ACCELERATED STRAIGHT-LINE ACCELERATED STRAIGHT-LINE

MOTIONMOTIONFORM TO USE WHEN

ACCELERATING OBJECT HAS AN INITIAL VELOCITY

FORM TO USE WHEN ACCELERATING OBJECT STARTS FROM REST.

tvvxfi )(

2

1

xavivf

222

)( tavvif

2)(2

1tatvx

i

tvxf )(

2

1

)( tavf

2)(2

1tax

xavf

22