Lecture: Acceleration A vector quantity

CCHS Physics

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Acceleration• The rate of change in velocity

• We can change the velocity of an object by:

– changing its speed– changing its direction– (or changing both)

aavg =ΔvΔt

Meaning of Acceleration• In everyday language ‘acceleration’ refers to

a gain in speed and ‘deceleration’ to a decrease in speed.

• In physics we will speak only of acceleration, and we will define it as being either positive or negative.

• Acceleration is positive when the change in velocity is positive and negative when the change in velocity is negative.

True or false?• If your speed is increasing, you must have a

positive acceleration.

ANSWER

False! • To increase speed in positive direction

= positive accelerationbut an increase in speed in the negative direction = negative acceleration

Acceleration Video

Reminders

• Velocity is the rate of change of position. It can be found graphically by taking the slope of a position vs. time graph.

• Acceleration is the rate of change of velocity. It can be found graphically by taking the slope of a velocity vs. time graph.

Let’s try some examples…• A car starts from rest and accelerates

uniformly at 3 m/s2 north. What is its velocity after 5 seconds?

• A bus traveling west at 20 m/s slows uniformly to 8 m/s in 6 seconds. What is its acceleration?

a =vf −v1

t 3=

vf −05

vf =15 m/s north

a =vf −v1

t a=

20 −86

a=−2 m/s2 =2 m/s2 east

Acceleration Graphing Movie

Try this one…

• During which of the following intervals is the acceleration the greatest: t = 0 - 3 st = 3 - 6.2 st = 6.2 - 9 s

• Ha ha that was a trick. It’s zero everywhere.

• How would a position vs. time graph look like if an object was accelerating?

Let’s Add in Some Calc!• Instantaneous Acceleration

• Relating back to position:

• So, acceleration is the second derivative of position with respect to time

a =dvdt

a =dvdt

=ddt

dxdt

⎛⎝⎜

⎞⎠⎟=

d2xdt2

Calc Example• If the velocity of dog is given by the

equation v(t) = 5t + 1, what is the acceleration of the dog at 4 s?

• If the position of a bee is given by the equation x(t) = .6t2 + 3t + 1, what is the acceleration of the snail at 7 s?

a =dvdt

=ddx

5t+1( ) = 5 m/s2

v =dxdt

=ddx

.6t2 + 3t+1( ) =1.2t+ 3

a =dvdt

=ddx

1.2t+ 3( ) =1.2 m/s2

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“What about Integrals?” - Integral Man

• So, remember that derivatives and integrals and opposites:

• EXAMPLE: If the acceleration of a bus is given by a(t) = 2t, what is the velocity after 4 s if the initial velocity is 7 m/s?

– We’ve got to solve for C using initial conditions

v = a( )dt∫ x = a( )dt∫∫

v = adt=∫ 2tdt0

4

∫ =t2 +C

v 0( ) =7 ⇒ 7 =02 +C ⇒ C =7 m/sv t( ) =t2 + 7

v 4( ) =42 + 7 = 23 m/s