Let’s do a quick recap of what we know at this point

• We can drive at one speed in our car

• This is constant speed• This occurs when we

have our cruise control activated

• Constant speed (velocity) equation

• Distance = speed*time

We created a distance-time graph

This is also called a position-time graph

• From this graph we used y = mx + b

• Xf = v(t) + Xi

• Slope is the speed or velocity

• The intercept is the initial position. Often this is at a reference point or ‘0’. However, our object may have a head start!

Time unit on the x axisDistance unit on the y axis

But there are many objects whose speed changes with respect to time

• This is an acceleration

Acceleration occurs when speed changes

• Now we don’t want our cruise control on

• We want to use our gas pedal or brake to make our car go faster or slower.

• We could also coast and that would make our car go slower and slower.

We can plot our velocity changes on a velocity-time graph (speed-time)

Speed-time for an object gaining 10 m/s each second (getting faster)

When we get a line on this graph, we have a constant acceleration

• Slope of this line is the acceleration

• How fast equation• Vf = Vi + a(t)• Y intercept is the initial

speed. Often this is zero but not always

So we are getting faster

• This is obvious to you drivers, right? You can press on the gas and get the car to go faster and faster.

• We now know that the speed of the car can be found using the how fast equation

• Final speed based on:• Vf = Vi + a(t)

But what about ‘how far’ the car moves?

• It turns out we can develop an equation for this, too.

• This is the distance that the car moves

• On my fit and on your car, you have an odometer which can help us measure distances we drive

Odometer below the speedometer

If we made a distance-time graph now, it would appear curved

This shape is called a PARABOLA. We call this our ‘getting faster’ parabola because

the car is getting faster

What does this parabola mean?• My distance is INCREASING

over successive time intervals.

• For example, if my car was traveling at 60 mph constant speed, every every hour I would travel 60 miles.

• Now my distance doesn’t stay constant, it increases.

• In the first second, I go 5 meters. In the next second, I go 15 additional meters, etc.

Top opening parabola

• Complete equation of parabola

• Ax2 + Bx + C = final position

• In physics terms:• Xf = ½ (a)(t2) + Vi(t) + Xi

• Acceleration is 2 (‘A’ ∙coefficient)

• Use tangents

What about my car getting slower? This is acceleration, too, because my

speed is changing.• Let’s say now I am

traveling at an initial 50 units of speed (mph for example) and decrease my speed by 10 units every second

Look at our speed-time graph

Still a linear function• Now a negative slope or

negative acceleration• Y-intercept represents

the initial velocity• So, my how fast

equation is:• Vf = Vi + a(t) or

• Vf = 50 + (-10)(t)

Now a negative diagonal

What about how far?

Parabola of car getting slower

• Parabola, too• But it now has the

opposite curvature• See how it is different

What about ‘how far’

‘Getting slower’ parabola• Negative acceleration• Ax2 + Bx + C = final position• In physics terms:• Xf = ½ (a)(t2) + Vi(t) + Xi

• Acceleration is 2 (‘A’ ∙coefficient)

• For this object:• Xf = ½ (-10)(t2) + (50)(t) + 0• Use tangents

A bottom opening parabola

We go less and less distance over successive time intervals• In the first second, I go 45

meters• In the next second, I go 35

additional meters.• In the next second, I go 25

additional meters• In the next second, I go 15

additional meters• In the last second, I go 5

meters• Isn’t the tangent line at 5

seconds a horizontal? Tangent lines imply speed. This would mean my car has stopped at 5 seconds.

So what types of motion can we do?

• Now we have everything covered!

* Car staying at one speed (constant speed)• Car accelerating

• Getting faster• Getting slower

Using these two equations can help us:Xf = ½ (a)(t2) + Vi(t) + Xi

(how far)

Vf = Vi + a(t)(how fast)