10-2 Distance, Displacement, and Acceleration for Linear Motion

1. 10-2 Distance, Displacement, and Acceleration for Linear Motion By: Elder Calderon

2. Displacement and Velocity • As covered in previous sections the derivative of Displacement is Velocity : • From this you can denote that the anti-derivative of acceleration must therefore be Displacement, giving:

3. Displacement under a Velocity graph • Displacement is defined as the distance away from the origin. • in a Velocity graph x-axis is time and the y-axis is velocity.

4. Displacement is given by the product of Velocity and Time: D= V x T • This can be introduced into a velocity graph by multiplying the two axis, and finding the area under the curve.

5. The area under the graph is equal to the displacement as it multiplies velocity (y-axis) and time (x- axis) giving m/s*s=m Velocity (m/s) Time (s)

6. This same concept of area under the curve can be found using integrals, the sum of the values of the curve will give the area of the curve, therefore Displacement , where a is the starting time and b is the ending time.

7. Distance or Displacement? • Displacement is not always equal to Distance, and therefore distance . Distance is an increasing function, a person running in a circle will always be increasing the distance they have run. Displacement however increases and decreases, after running one lap in a circle displacement is 0.

8. This graph is a velocity vs time graph, and the integral of the graph gives only displacement, in order to give distance the graph must be all positive

9. Therefore Distance must be the absolute value of the area under the curve and can be written out as: . Velocity The previous portion that was under x-axis is now above and gives a positive area . This gives total distance traveled

10. Acceleration and Velocity • Velocity is distance traveled in a set amount of time, distance usually being in meters or feet and time being in one second. • The resulting units being M/S (meters/second). • Acceleration is the change in velocity in a set amount of time. • The units for Acceleration are: M/S*S

11. In an Acceleration graph the Y-axis is Acceleration (m/s*s) and the x-axis is time (s). • The y and x- axis can be multiplied to be velocity, (M/(s*s))*s=M/S (units for velocity) • Therefore the area or integral of an acceleration graph is velocity.

12. Therefore the area or the sum of the y values in an acceleration graph is equal to the velocity. The sum of the acceleration values from time a to time b is equal to the velocity at time b:

13. Recap • Displacement: • Distance: • Velocity: