Unit 7B

Integrals Part 2

Lesson 2: Displacement, Velocity, and Acceleration

A particle is moving in a straight line. The particle’s displacement from a fixed point is given by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»t«/mi»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»7«/mn»«mi»t«/mi»«mo»+«/mo»«mn»9«/mn»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/math».

a.
Determine the velocity and the acceleration of the particle at any time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».

b.
Determine the turning points of the particle.

c.
Describe the motion at times «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»3«/mn»«/math».

d.
Sketch and describe the following graphs.

i.
displacement «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» verses time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math»

ii.
velocity «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» verses time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math»

iii.
acceleration «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» verses time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math»

a.
To find velocity, find the derivative of the displacement function.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»3«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»7«/mn»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/math»

To find acceleration, find the derivative of the velocity function.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mn»6«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»10«/mn»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/math»

The velocity at any time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math» is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»7«/mn»«/math» and the acceleration at any time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math» is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mn»6«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»10«/mn»«/math».

b.
The turning points occur when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mn»0«/mn»«/math».

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»7«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»7«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced»«mrow»«mi»t«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mn»3«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left center center center center center right center left¨»«mtr»«mtd»«mi»t«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«mtd/»«mtd/»«mtd»«mi»and«/mi»«/mtd»«mtd/»«mtd/»«mtd»«mn»3«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«/mtd»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»

For «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»1«/mn»«/math»:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mn»1«/mn»«/mfenced»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mfenced»«mn»1«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»7«/mn»«mfenced»«mn»1«/mn»«/mfenced»«mo»+«/mo»«mn»9«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mo»§#8722;«/mo»«mn»5«/mn»«mo»+«/mo»«mn»7«/mn»«mo»+«/mo»«mn»9«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»12«/mn»«/mtd»«/mtr»«/mtable»«/math»

For «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«/math»:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»s«/mi»«mfenced»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«/mfenced»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mfenced»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»7«/mn»«mfenced»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«/mfenced»«mo»+«/mo»«mn»9«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»343«/mn»«mn»27«/mn»«/mfrac»«mo»§#8722;«/mo»«mfrac»«mn»245«/mn»«mn»9«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mn»49«/mn»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mn»9«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»343«/mn»«mn»27«/mn»«/mfrac»«mo»§#8722;«/mo»«mfrac»«mn»735«/mn»«mn»27«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mn»441«/mn»«mn»27«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mn»243«/mn»«mn»27«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»292«/mn»«mn»27«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
 
The turning points occur at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»12«/mn»«/mrow»«/mfenced»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»292«/mn»«mn»27«/mn»«/mfrac»«/mrow»«/mfenced»«/math».

c.
Create a table of values for the various values of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»7«/mn»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»7«/mn»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»1«/mn»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»7«/mn»«mn»3«/mn»«/mfrac»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«/math»

The motion of the particle:

  • At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»0«/mn»«/math», the particle is moving in the positive direction (to the right or up).
  • At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»1«/mn»«/math», the particle is stopped.
  • At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»2«/mn»«/math», the particle is moving in the negative direction (it is moving back toward the starting point – to the left or down).
  • At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»2«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math», the particle is stopped.
  • At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»3«/mn»«/math», the particle is moving in the positive direction

d.
A table of values may be used to assist in sketching each of the graphs.

i.
The following graph shows the displacement of the particle.
 



The particle started nine units from the origin, and then moved three units in the positive direction. Then, the particle reversed direction and moved toward the origin. After «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»2«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math», the particle reversed direction again and moved in the positive direction, away from the origin.

ii.
The following graph shows the velocity of the particle.



 
The velocity is decreasing from «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»0«/mn»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»1«/mn»«/math». When «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»1«/mn»«/math», the particle stops (the velocity is zero). From «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo»§#60;«/mo»«mi»t«/mi»«mo»§#60;«/mo»«mn»2«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math», the particle is moving in the opposite direction. The particle stops again at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»2«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math». After that, the particle turns around and the velocity increases as the particle moves in a positive direction.

iii.
The following graph shows the acceleration of the particle.




The particle’s acceleration is increasing from negative to positive at a constant rate. When «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is negative, the particle’s velocity is decreasing as long as velocity positive. When «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is positive, the particle’s velocity is increasing as long as velocity positive.
Recall when finding the derivative of a polynomial function, the degree of the function is decreased by one. As seen in this Example 2, the displacement function was a cubic function, the velocity function was a quadratic function, and the acceleration function was a linear function.