1. Kinematic Equations for One Dimensional Motion Under Constant Acceleration
Explain how Ms. Bjork’s motion is classified as 1D under constant acceleration
Ms. Bjork

2. Velocity as a function of time
If acceleration is constant, the average a over any time interval also equals the instantaneous a at any instant within that interval
But we can replace ā with a, and we can say that ti=0, giving us:
Or:
We already have an equation for average acceleration:

3. Displacement as a function of velocity and time
We know that:
and:
Giving us:
Average velocity can be expressed both as the change in position over change in time, and also as the arithmetic mean of the initial and final values of the velocity
Which is an equation the displacement of an object as a function of velocity and time

4. Displacement as a function of time
Now we take our last equation and our equation for velocity as a function of time to get:
We can now check this using the calculus:
First we rewrite our equation as:
Now we differentiate with respect to time:

5. Velocity as a function of displacement
To get velocity as a function of displacement, we need to get rid of time:
We have:
and:
So we substitute in this value of t:
Giving us:

6. Kinematic Equations
Motion in a straight line under constant acceleration:
V as a function of t
X as a function of v and t
X as a function of t
V as a function of x

7. Kinematic equations from the calculus

8. The area under the curve of a velocity vs time graph
Displacement
The area under the curve of a velocity vs time graph
For this graph of v vs t the time interval tf –ti has been divided into many small intervals (each of Δtn)
Looking at one of these rectangles we see that the area would be vΔtn
So: A=Δxn= vΔtn (meaning that this area gives us the displacement during this small time interval)

9. Displacement
The area under the curve of a velocity vs time graph
If we add up all of these rectangles for the total time interval of tf – ti we then get the total displacement:
If we make these time intervals smaller and smaller (n∞, or Δtn0) the value of the area we are finding gets closer and closer to the actual area under the curve
Displacement = area under the v vs t graph

10. Displacement This limit is also known as a definite integral:
The area under the curve of a velocity vs time graph
This limit is also known as a definite integral:

11. Displacement
The area under the curve of a velocity vs time graph
Sometimes this area under the graph is much simpler to analyze:
This graph shows an object with a constant velocity. The area under this graph is easy to find (bxh) and it gives the displacement over this time interval as well.
The cool thing is that this always works, if you are given a graph of v vs t you can find Δx by finding the area under the curve.

12. Velocity
The area under the curve of an acceleration vs time graph
Just as the area under the curve of a v vs t graph gives Δx, the area under the curve of an a vs t graph gives Δv.
Think about the unit analysis. Consider this graph of a vs t, if you find the area under the curve this gives you:
ΔaΔt (m/s2)s = m/s (velocity) ax
ax=axi=constant
axi
axi

13. Kinematic equations from calculus
We know acceleration is defined as:
We take the integral (antiderivative) to get:
When a is constant it can be pulled from the integral to get:
The value of C depends on the initial conditions of the motion. We let vx
= vxi and t = 0 to get:
Now we let vx = vxf after the time interval t has passed and substitute the value found for C into the equation cv cv cv

14. Kinematic equations from calculuscv
We know velocity is defined as:
We take the integral (antiderivative) to get:
We know that here vx=vxf=vxi+axt
When vxi and a are constants they can be pulled from the integral to get:
The value of C depends on the initial conditions of the motion. We let x= xi and t = 0 to get:
Now we let x = xf after the time interval t has passed and substitute the value found for C into the equation cv cv cv cv cv
cvcv cv