Kinematics, getting position and velocity equations from an acceleration graph

In summary, the conversation involves a question about a car's acceleration and braking on a road, and the task of creating graphs and equations to represent the car's velocity and position over time. The individual also provides their attempted solution, including expressions for velocity and position and calculations for the distance the car stops. They ask for help in identifying their mistake and provide a request for a plot of acceleration vs time.
  • #1
Telemachus
835
30

Homework Statement


I have some doubts to the next exercise. It reads:

A car that at the initial time at rest was 2 km of a post roadman is subjected to acceleration, whose tangential component during the first 10 seconds, varies with time as shown in Fig. From that moment the brakes are applied to subject the car to an acceleration whose tangential component is constant and stops the car in the first 100 meters

index.php?action=dlattach;topic=36734.png


a) Complete the graph shown qualitatively in Fig.
b) To obtain expressions in terms of time for the module of the velocity vector of the car, valid for each of the time intervals considered.
c) To obtain expressions in terms of time to measure the position of the car along the road, about the position roadman, valid for each of the time intervals considered.
d) Determine how far the post roadman stops the car.
e) Perform qualitative graphical function of time for the module of its velocity vector and its position, measured along the path, for the post runners.

The Attempt at a Solution



At first I did was plotted, as requested. Consider that the braking acceleration is constant, and therefore a constant line plot from the 10s.

Then I got the expressions for the velocity (from here m=meters, t=time in seconds):

[tex]a(t)=\displaystyle\frac{tm}{5s^3}[/tex] [tex]t\leq{10}[/tex]

[tex]v(t)=\displaystyle\frac{t^2m}{10s^3}[/tex] [tex]t\leq{10}[/tex]

[tex]v(10s)=10m/s[/tex]

From here I wanted to get acceleration after ten seconds:

[tex]\displaystyle\frac{\Delta x}{\Delta v}=\Delta t\Rightarrow{\Delta t=10s}[/tex]

[tex]a=\displaystyle\frac{\Delta v}{\Delta t}=\displaystyle\frac{-1m}{s^2}[/tex]

[tex]v(t)=20\displaystyle\frac{m}{s}-\displaystyle\frac{tm}{s^2}[/tex]

b) [tex]v(t)=\begin{Bmatrix} \displaystyle\frac{t^2m}{10s^3} & \mbox{ si }& t\leq{10}\\20m/s-tm/s^2 & \mbox{si}& 20s>t>10s\end{matrix}[/tex]

c)Position [tex]x_i=2000m[/tex]

Integrating I get
[tex]x(t)=2000m+t^3m/30s^3 \forall{t\in{[0,10]}}[/tex]

For the initial position in the second interval I've considered the position at 10s:
[tex]x(10)=2000m+1000m/30=\displaystyle\frac{6100m}{3}\approx{2033.3m}[/tex]

Then integrating:

[tex]x(t)=\begin{Bmatrix} x(t)=2000m+t^3m/30s^3 \forall{t\in{[0,10]}}\\\displaystyle\frac{6100m}{3}+20\displaystyle\frac{m}{s}-\displaystyle\frac{t^2m}{2s^2}\forall{t\in{[10,20]}\end{matrix}[/tex]

This is where the problem is, the point d asks to calculate the distance that the vehicle is stopped
This distance should be according to my calculations:

[tex]\displaystyle\frac{6100m}{3}+100m=\displaystyle\frac{6400m}{3}[/tex]

But this should be equal to x (t), evaluated at the 20s:

[tex]x(20s)=\displaystyle\frac{6100m}{3}+\displaystyle\frac{20m}{s}20s-\displaystyle\frac{(20s)^2m}{2s^2}=\displaystyle\frac{6100}{3}m+400m-200m=\displaystyle\frac{6700m}{3}\approx{2233.3m}[/tex]

What Am I doing wrong?

Bye there!
 
Last edited:
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  • #2
Can you show your plot of acceleration vs time, please?

ehild
 

Related to Kinematics, getting position and velocity equations from an acceleration graph

1. How do you find the position equation from an acceleration graph?

The position equation can be found by taking the integral of the acceleration graph. This will result in the velocity equation, which can then be integrated again to obtain the position equation.

2. Can you explain the process of getting velocity equations from an acceleration graph?

To get the velocity equation from an acceleration graph, you need to take the integral of the graph. This will give you the equation for velocity over time. You can also find the average velocity by finding the area under the curve of the acceleration graph.

3. Is it possible to determine the acceleration from a position equation?

Yes, it is possible to determine the acceleration from a position equation. You can find the velocity equation from the position equation by taking the derivative. Then, you can find the acceleration by taking the derivative once more.

4. Can you use an acceleration graph to predict an object's position and velocity at a specific time?

Yes, an acceleration graph can be used to predict an object's position and velocity at a specific time. By analyzing the graph and using the equations derived from it, you can determine the position and velocity at any given time.

5. How does kinematics play a role in understanding the motion of objects?

Kinematics is the study of motion, specifically the mathematical description of an object's position, velocity, and acceleration over time. By understanding the principles of kinematics, scientists can accurately predict the motion of objects and study their behavior in various situations.

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