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0.4 Linear equations  (Page 2/3)

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A 1 = π r 1 2 = π ( 4 . 00 cm ) 2 size 12{A rSub { size 8{1} } =π`r rSub { size 8{1} rSup { size 8{2} } } =π` \( 4 "." "00"` ital "cm" \) rSup { size 8{2} } `} {}
A 2 = π r 2 2 = π ( 2 . 50 cm ) 2 size 12{A rSub { size 8{2} } =π`r rSub { size 8{2} rSup { size 8{2} } } =π \( 2 "." "50"` ital "cm" \) rSup { size 8{2} } } {}

We can apply equation (4) to find the velocity of the fluid in Pipe 2. The steps are shown below

v 2 = π ( 4 . 00 cm ) 2 π ( 2 . 50 cm ) 2 ( v 1 ) size 12{v rSub { size 8{2} } = left ( { {π` \( 4 "." "00"` ital "cm" \) rSup { size 8{2} } } over {π` \( 2 "." "50"` ital "cm" \) rSup { size 8{2} } } } right )` \( v rSub { size 8{1} } \) } {}
v 2 = 16 . 00 6 . 25 ( v 1 ) size 12{v rSub { size 8{2} } = { {"16" "." "00"} over {6 "." "25"} } ` \( v rSub { size 8{1} } \) } {}
v 2 = 2 . 56 v 1 size 12{v rSub { size 8{2} } =2 "." "56"`v rSub { size 8{1} } } {}
v 2 = 2 . 56 × 3 . 00 m / s size 12{v rSub { size 8{2} } =2 "." "56" times 3 "." "00"`m/s} {}
v 2 = 7 . 68 m / s size 12{v rSub { size 8{2} } =7 "." "68"`m/s} {}

If we compare our result for the velocity of the fluid in Pipe 2 with the velocity of fluid in Pipe 1, we immediately see that the velocity of the fluid increases as it moves from a pipe with a larger cross-sectional area to another pipe with a smaller cross-sectional area. This result is intuitive with what we may have observed through personal experience.

Question: Make a plot that relates the dependent variable ( v 2 ) and the independent variable ( v 1 ).

Referring to equation (10), we note that a linear relationship exists between v 2 and v 1 . For the straight line, the slope is 2.56 and the y-intercept is 0. We plot the line below

Graph of the linear relationship between velocities.

It is important to note that each axis is labeled and includes the units associated with each variable.

Mechanics – velocity and acceleration

Let us consider a car that is traveling at an unknown initial velocity ( v 0 ). The driver of the car decides to enter the on-ramp of a freeway. The driver knows that he will need to increase his velocity in order blend in with the other traffic on the freeway. While situated in the on-ramp, the driver of the car applies pressure to the accelerator of the car. Let us consider that this action occurs at a specific instant of time ( t 0 ). By applying constant pressure on the accelerator, the driver causes the car to accelerate at a constant acceleration ( a ).

This constant acceleration causes the velocity of the car to increase. During the time interval that the driver applies constant pressure to the accelerator, the velocity of the car can be expressed as a function of time

v ( t ) = a t + v 0 size 12{v \( t \) =a`t+v rSub { size 8{0} } } {}

Inspection of this equation reveals that the velocity is a linear function of time. Here the dependent variable would be velocity and the independent variable would be time. The slope of the straight line that is associated with the equation would equal to the acceleration ( a ). The y-intercept of the equation would be the initial velocity ( v 0 ).

Let us apply what we have learned about the relationship between velocity and acceleration coupled with our knowledge of linear equations to work a problem.

Question: At an instant of time ( t 1 = 1.00 s) after depressing the accerlerator, the driver observes that the car is traveling at a velocity ( v 1 = 17.00 m/s). At an instant of time one second later (that is at t 2 = 2.00 s), the driver observes that the velocity of the car has increased to a value ( v 2 = 22.0 m/s). Determine the initial velocity of the vehicle and the value for the constant acceleration ( a ).

Solution: We know the values of the velocity at two instants of time, 1.00 seconds and 2.00 seconds. Because the acceleration is constant, we also know that the relationship between velocity and time is linear.

The slope of the line that relates velocity to time is equal to the acceleration ( a ) and the y-intercept corresponds to the intial velocity ( v 0 ).
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Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
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