Deriving an equation for displacement and acceleration (given velocity)

[southernbelle]

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?

 

[minger]

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 into get your constant.

 

[tiny-tim]

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!

x(0)=0, so t²/2 + 3t + C has to be 0 when t = 0, so C = … ?

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.

 

[HallsofIvy]

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.

 

[southernbelle]

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.
? :(

 

[minger]

Well it's not slope intercept form because it's not a simple linear equation. Graphing these is quite easy. Time is your independent 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.