Deriving an equation for displacement and acceleration (given velocity)

  • #1

Homework Statement


For 0<t<1, v(t) = t +3
For 1<t<2, v(t) = 5-t
Assume x(o)=0

A) Draw corresponding displacement and acceleration diagrams.
B) Determine the equation for each segment


Homework Equations


Acceleration is the derivative of velocity.
Velocity is the derivative of displacement.


The Attempt at a Solution


I can draw the acceleration diagram and write the equation so no problem there.
My problem is drawing the displacement diagram.

I got the equations for displacement. They are:
For 0<t<1
t2/2 + 3t + C
For 1<t<2
5t - t2/2 + C

I cannot figure out how to evaluate that constant and plot that on a graph. Also, my teacher mentioned finding the area under the original curve and plotting that. The area = 3.5but that's not a point to plot. What do I dO?
 

Answers and Replies

  • #2
minger
Science Advisor
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It seems to me that you have everything except the constants of integration. You are told to assume that x(0) = 0, meaning that at time of zero, you have no displacement. Plug it in to get your constant.
 
  • #3
tiny-tim
Science Advisor
Homework Helper
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For 0<t<1, v(t) = t +3
For 1<t<2, v(t) = 5-t
Assume x(o)=0

I got the equations for displacement. They are:
For 0<t<1
t2/2 + 3t + C
For 1<t<2
5t - t2/2 + C

I cannot figure out how to evaluate that constant and plot that on a graph.
Hi southernbelle! :smile:

x(0)=0, so t2/2 + 3t + C has to be 0 when t = 0, so C = … ? :wink:
Also, my teacher mentioned finding the area under the original curve and plotting that. The area = 3.5but that's not a point to plot. What do I dO?
ah … your 3.5 is just the area for t = 1 …

your teacher meant the area A(t) up to time t for any t

plot A(t) against t, and that's the displacement. :smile:
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,833
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You have
For 0<t<1
x(t)= t2/2 + 3t + C
For 1<t<2
x(t)=5t - t2/2 + C
and x(0)= 0.

Be careful- the two "C"s are necessarily the same.

Use x(0)= 0 to find C in the first equation. Then use the fact that the two equations must give the same result at x= 1 to find C in the second equation.
 
  • #5
Okay, so

I evaluated the constants.
For the first equation:
C = 0
For the second equation
C = -1

But how do I plot those? Would I use the coordinates (0,0) and (1, -1) ?

I am thinking that the Constant is where you start on the y-axis and then you use the slope to go from there.

But the equation is not written in slope intercept form.
? :(
 
  • #6
minger
Science Advisor
1,495
2
Well it's not slope intercept form because it's not a simple linear equation. Graphing these is quite easy. Time is your independant variable, it depends on nothing, so it's your x-axis. The velocity/disp/accel are dependent on time x = f(t), so it's your y-axis. Just start at t=0, plug it into your equation and put a point, then go to 0.1, or whatever you choose, and calculate x. Rinse and repeat until you get to time = 1.0 seconds, then switch to the other equation.
 

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