Unit 7B

Integrals Part 2

Lesson 2: Displacement, Velocity, and Acceleration

Acceleration

A car is travelling at a velocity of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»15«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math». The truck travelling in front of the car is driving at a velocity of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math». The driver of the car steps on the gas to increase the velocity of the car to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»22«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math» in order to pass the truck. By increasing its speed, the car has accelerated.

Acceleration is the rate at which velocity changes per unit of time. If the car’s velocity increased from «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»15«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»22«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math» in four seconds, the acceleration is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»22«/mn»«mo»§#8722;«/mo»«mn»15«/mn»«/mrow»«mn»4«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«mo»=«/mo»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«msup»«mi mathvariant=¨normal¨»s«/mi»«mn»2«/mn»«/msup»«/math». Acceleration is the rate of change of velocity.

     
When «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#916;«/mo»«mi»t«/mi»«/math» approaches zero, the average acceleration approaches the acceleration at a given moment in time. This is called instantaneous acceleration, which is denoted by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math».

Therefore, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mo»§#916;«/mo»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mo»§#916;«/mo»«mi»v«/mi»«/mrow»«mrow»«mo»§#916;«/mo»«mi»t«/mi»«/mrow»«/mfrac»«/math».

Since «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mo»§#916;«/mo»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mo»§#916;«/mo»«mi»v«/mi»«/mrow»«mrow»«mo»§#916;«/mo»«mi»t«/mi»«/mrow»«/mfrac»«/math» is the derivative of the velocity function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«/math» with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math», «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»v«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».

Recall the derivative of displacement is velocity, that is «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»«/math».

Therefore, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«/math», which is the second derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«/math» with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».

Therefore, «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»«mfrac»«mrow»«mi»d«/mi»«mi»v«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».

If «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«/math» is measured in metres and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math» is measured in seconds, the units for acceleration are «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«msup»«mi mathvariant=¨normal¨»s«/mi»«mn»2«/mn»«/msup»«/math».

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»velocity«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mi»s«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»acceleration«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mi»v«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»t«/mi»«/mfenced»«mo mathvariant=¨italic¨»=«/mo»«mfrac»«msup»«mi mathvariant=¨italic¨»d«/mi»«mn mathvariant=¨italic¨»2«/mn»«/msup»«mrow»«mi mathvariant=¨italic¨»d«/mi»«msup»«mi mathvariant=¨italic¨»t«/mi»«mn mathvariant=¨italic¨»2«/mn»«/msup»«/mrow»«/mfrac»«mi»s«/mi»«mfenced»«mi mathvariant=¨italic¨»t«/mi»«/mfenced»«mo mathvariant=¨italic¨»=«/mo»«mi»s«/mi»«mo mathvariant=¨italic¨»``«/mo»«mfenced»«mi mathvariant=¨italic¨»t«/mi»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»