1. 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

2. We created a distance-time graph
This is also called a position-time graph
Time unit on the x axis
Distance unit on the y axis
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!

3. But there are many objects whose speed changes with respect to time
This is an acceleration

4. 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.

5. We can plot our velocity changes on a velocity-time graph (speed-time)
When we get a line on this graph, we have a constant acceleration
Speed-time for an object gaining 10 m/s each second (getting faster)
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

6. 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)

7. 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

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

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

10. 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.

11. 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

12. What about my car getting slower
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

13. Look at our speed-time graph
Still a linear function
Now a negative diagonal
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)

14. Parabola of car getting slower
What about how far?
Parabola of car getting slower
Parabola, too
But it now has the opposite curvature
See how it is different

15. What about ‘how far’ ‘Getting slower’ parabola
A bottom opening 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

16. 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.

17. So what types of motion can we do?
Now we have everything covered!
So what types of motion can we do?
* 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)