How to find acceleration or displacement with respect to time?

In summary, the conversation discusses finding acceleration or displacement with respect to time using the definition of acceleration and writing it as an ODE. It is mentioned that depending on the sign of the constant k, the solution for x(t) will involve a linear combination of either real or complex exponentials. The conversation also mentions setting up a characteristic equation to solve for the nonhomogeneous part of the ODE.
  • #1
sid_galt
502
1
[tex]
a(x)= kx + 1
[/tex]
where [tex]a(x)[/tex] is acceleration with respect to displacement along the x-axis and [tex]x[/tex] is the displacement itself while k is the constant.

How to find acceleration or displacement with respect to time?
 
Last edited:
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  • #2
Use the definition of acceleration to write the ODE.Depending on that sign of "k",the answer for x(t) is a linear comb.of complex or real exponentials...

Daniel.
 
  • #3
Plus a constant (the ODE is nonhomogenous).

Daniel.
 
  • #4
I can't figure out the ODE. Can you help me?
 
  • #5
sid_galt said:
[tex]
a(x)= kx + 1
[/tex]
where [tex]a(x)[/tex] is acceleration with respect to displacement along the x-axis and [tex]x[/tex] is the displacement itself while k is the constant.

How to find acceleration or displacement with respect to time?

acceleration is the second derivative of x wtr to time :

[tex] \ddot{x} = kx + 1[/tex]

[tex]\ddot{x} -kx -1 = 0[/tex]

Can you solve it from here. You will need to set up the associated caracteristic equation and based upon the sign of k you will get a linear combination of exponentials or complex exponentials

marlon
 
  • #6
The nonhomogeneity part is usually left in the RHS...

Daniel.
 

Related to How to find acceleration or displacement with respect to time?

1. How do I calculate acceleration?

To calculate acceleration, you need to know the change in velocity and the time it took for that change to occur. The formula for acceleration is a = (vf - vi) / t, where a is acceleration, vf is final velocity, vi is initial velocity, and t is time.

2. How is acceleration related to displacement?

Acceleration is the rate of change of velocity over time. It is directly related to displacement, as a change in acceleration will result in a change in displacement. The formula for displacement is d = v0t + 1/2at^2, where d is displacement, v0 is initial velocity, a is acceleration, and t is time.

3. What unit is used to measure acceleration?

The SI unit for acceleration is meters per second squared (m/s^2). This means that for every second, the velocity changes by the specified amount in meters per second.

4. How can I determine the acceleration from a velocity-time graph?

The slope of a velocity-time graph represents the acceleration. The steeper the slope, the greater the acceleration. You can find the slope by selecting two points on the graph and using the formula m = (y2 - y1) / (x2 - x1), where m is the slope, x1 and x2 are the time values, and y1 and y2 are the velocity values.

5. How can I find the displacement using an acceleration-time graph?

The area under the curve on an acceleration-time graph represents the displacement. To find the displacement, you can calculate the area of the trapezoid formed by the graph using the formula A = 1/2(a1 + a2) * t, where A is the area, a1 and a2 are the two acceleration values, and t is the time interval.

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