Acceleration

Acceleration is a vector quantity which is defined as the rate at which an object

changes its velocity. An object is accelerating if it is changing its velocity.

What does acceleration really mean?

• 4.0 m or 4.0 m/s every second s2

Every second the object is increasing its speed 4.0 m s

-4.0 m s 2

Every second the object is decreasing its speed by -4.0 m s

Acceleration due to Gravity

There is also gravitational acceleration of

-9.8 m/s2 on all objects thrown in the air

• Now consider a car moving with a rightward (+), changing velocity - that is, a car that is moving rightward but speeding up or accelerating

The average accelerationaverage acceleration of an object is defined to be the ratioratio of its change in velocitychange in velocity to the

time taken to change the velocitytime taken to change the velocity.

aaavavaaavav = average acceleration;

in units of m/s/s, m/sm/s/s, m/s22

= vv

vv = change in velocity; in units of m/s, km/hrm/s, km/hr

tt

tt = change in time; in units of s, min, hrs, min, hr, etc...

A positive signpositive sign indicates that the velocity is increasingincreasing. It will also be an increaseincrease ininspeedspeed if the object is traveling in the positive positive directiondirection. It is a decrease in speed otherwise.

A negative signnegative sign indicates that the velocityis decreasingdecreasing. It will also be a decrease indecrease inspeedspeed if the object is traveling in the positive positive directiondirection. It is an increase in speed otherwise.

The “The “signsign” of the acceleration” of the accelerationindicates whether the velocity isindicates whether the velocity is

increasingincreasing or or decreasingdecreasing..

It is important to note that information about It is important to note that information about an an object’sobject’s

acceleration tells us how the object’s velocity is acceleration tells us how the object’s velocity is changing.changing.

In order In order to know what this change in velocityto know what this change in velocityis doing to the object’s speedis doing to the object’s speed, we , we mustmust

know the know the directiondirection the object is traveling the object is traveling..

As a rule, if the As a rule, if the object’s velocity and accelerationobject’s velocity and accelerationare in the same directionare in the same direction (have the same sign), (have the same sign),we can say that the object’s we can say that the object’s speed is increasingspeed is increasing..

If the If the velocity and acceleration are in oppositevelocity and acceleration are in oppositedirectionsdirections (have opposite signs), we know (have opposite signs), we know

that the object’s that the object’s speed is decreasingspeed is decreasing..

1. According to the information above, what is the average acceleration of the racer in m/s2? Record your answer to the nearest hundredth.

1. According to the information above, what is the average acceleration of the racer in m/s2? Record your answer to the nearest hundredth.

a = vf – vi = 4.5 m/s – 2.2 m/s = .46 m/s2

tf – ti 5.0s – 0.0 s

• 2. A rock starts rolling down a hill at

.0194 m/s. One minute later it is rolling

at .244 m/s. What is its acceleration?

• 2. A rock starts rolling down a hill at

.0194 m/s. One minute later it is rolling at

.244 m/s. What is its acceleration?

a = vf – vi =.244 m/s - .0194 m/s =.00374 m/s2

∆t 1min 60.s

1min

• 3. A box accidentally falls from a truck traveling 20.1 m/s and decelerates at

-6.627 m/s2 before it comes to a stop. Find the time it takes it to stop.

• 3. A box accidentally falls from a truck traveling 20.1 m/s and decelerates at

-6.627 m/s2 before it comes to a stop. Find the time it takes it to stop.

a = vf – vi so ∆t = vf – vi = 0-20.1m/s

∆t a -6.627 m/s2

∆t = 3.03 s

• 4. A rocket starting from rest undergoes an acceleration of 0.60 m/s2 straight upward for

3.0 minutes, when the fuel gives out. What is its final velocity at this point?

• 4. A rocket starting from rest undergoes an acceleration of 0.600 m/s2 straight upward for 3.00 minutes, when the fuel gives out. What is its final velocity at this point?

a = vf – vi so vf – vi = a ∆t

∆t vf = vi + a ∆t

vf = 0 m/s + 0.600 m/s2 (3.00 min 60 s) 1 min

vf = 108 m/s

Constant Acceleration Equations

• vf = vi + a∆t

• a = vf-vi = ∆v tf-ti ∆t

• ∆x = vavg ∆t or ∆x = (vf + vi) ∆t 2 • ∆x = vi ∆t + ½ a ∆t

ConstantConstant

vvff = v = vii + at + atx = vx = vavavtt

vvavav = (v = (vff + v + vii)/2)/2

x = vx = viit + 1/2att + 1/2at22

vvff22 = v = vii

22 + 2ax + 2ax

t = timex = displacement

vi = initial velocityvav = average

velocitya = acceleration

vf = final velocity

Position – Time Graphs SummarizedPosition – Time Graphs Summarized

• the the y-coordinatey-coordinate at any time gives the at any time gives the positionposition of the object of the object

• the the slopeslope of a position-time graph at any instant is the instantaneous of a position-time graph at any instant is the instantaneous velocityvelocity of the object of the object

• horizontalhorizontal graph segments indicate that the object is “ graph segments indicate that the object is “at restat rest””

• graph segments moving graph segments moving upwardupward imply movement in the imply movement in the positive positive directiondirection

• graph segments moving graph segments moving downwarddownward imply movement in the imply movement in the negative negative direction direction

• straightstraight line graph segments indicate line graph segments indicate constant speedconstant speed

• curvingcurving graph segments indicate graph segments indicate changing speedchanging speed

• graph segments becoming graph segments becoming steepersteeper indicate an indicate an increase in speedincrease in speed

• graph segments becoming graph segments becoming less steepless steep indicate a indicate a decrease in speeddecrease in speed

• a a change of directionchange of direction is indicated whenever the graph is is indicated whenever the graph is concaveconcave upward or downwardupward or downward

Velocity – Time Graphs SummarizedVelocity – Time Graphs Summarized

• the the y-coordinatey-coordinate at any time gives the at any time gives the velocityvelocity of the object of the object• the the slopeslope of a velocity-time graph is the of a velocity-time graph is the accelerationacceleration of the object of the object• horizontalhorizontal graph segments indicate that the object has graph segments indicate that the object has constant constant velocityvelocity• graph segments graph segments above the x-axisabove the x-axis imply movement in the imply movement in the positive positive directiondirection• graph segments graph segments below the x-axisbelow the x-axis imply movement in the imply movement in the negative negative direction direction • horizontal segments horizontal segments on the x-axison the x-axis indicate no movement indicate no movement• straightstraight line graph segments indicate line graph segments indicate constant accelerationconstant acceleration• graph segments moving upward indicate an graph segments moving upward indicate an increase in velocityincrease in velocity• graph segments moving downward indicate a graph segments moving downward indicate a decrease in velocitydecrease in velocity• a a change of directionchange of direction is indicated whenever the graph crosses the x- is indicated whenever the graph crosses the x- axisaxis• an an increase in speedincrease in speed is indicated by graph segments moving is indicated by graph segments moving away away from the x-axisfrom the x-axis

Acceleration – Time Graphs SummarizedAcceleration – Time Graphs Summarized

• the the y-coordinatey-coordinate at any time gives the at any time gives the accelerationacceleration of the object of the object• horizontalhorizontal graph segments indicate that the object has graph segments indicate that the object has constant constant accelerationacceleration• a horizontal graph segment a horizontal graph segment on the x-axison the x-axis indicates that the object indicates that the object has has constant velocityconstant velocity (no acceleration) (no acceleration)• graph segments graph segments above the x-axisabove the x-axis imply imply increasing velocities increasing velocities • graph segments graph segments below the x-axisbelow the x-axis imply imply decreasing velocities decreasing velocities • no changes in direction may be inferred from these graphsno changes in direction may be inferred from these graphs

At the introductory physics level, we typically only deal At the introductory physics level, we typically only deal with constant acceleration situations, so acceleration with constant acceleration situations, so acceleration graphs generally consist of horizontal segments only.graphs generally consist of horizontal segments only.