Calculate displacement when acceleration is dependent on displacement?

In summary, the person is trying to calculate the kinetic energy of an object being accelerated from rest. They note that there is a shortcut for the math involved, which is conservation of energy. They say that the person they are talking to is saying the same thing.
  • #1
Jasoni22
8
0
I really have no idea where to start with this.

I have [itex]a=\frac{2.071\cdot10^{11}A}{mx^{2}}[/itex] m/s2, where [itex]A[/itex] and [itex]m[/itex] are constants. Since the acceleration is dependent on position, and the position is obviously dependent on acceleration, I don't know how to separate the two in order to do any calculations.

Basically, what I'm trying to do is for any initial position [itex]x_{0}[/itex], calculate the position at time [itex]t[/itex] of an object being accelerated from rest by a force that is decreasing as the square of the distance from the object.
 
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  • #2
Hi.

You should solve the differential equaitond^2 x/dt^2 = C x^-2

Regards.
 
  • #3
An alternative approach that is often useful when you have a(x) is to note that a=v*dv/dx = 1/2*dv2/dx. Equating that to a(x) and integrating over dx is the source of the concept of kinetic energy and its connection to potential energy, for example. Solving an equation like 1/2*dv2/dx = a(x) is then a first-order differential equation in x, rather than second order in t. You can often get v(x), which is equivalent to calculating the change in potential energy and attributing the change in kinetic energy to it. Once you have v(x), you write v(x) = dx/dt and solve the first-order differential equation in t to get x(t), if all the functions are integrable and invertiable as needed. This way you only have to deal with first-order differential equations that can be solved by integration (if you are lucky). Or, you can just do it sweet_springs' way, if you can see by inspection the function you need. I mention the other way because of its important connection to the concept of conservation of energy, which is merely a shortcut for the math I described.
 
  • #4
Hi. We are saying the same thing.

2v dv/dt = 2vC x^-2

d (v^2) = 2C x^-2 dx

v^2 = -2C /x + c

v^2 + 2C /x = v0^2 + 2C /x0

Regards.
 
  • #5


I understand your confusion with this problem. The relationship between acceleration and displacement is not a simple one and can be difficult to calculate. However, there are a few steps we can take to try to approach this problem.

First, we need to understand the relationship between acceleration and displacement in this scenario. From the given information, we know that the acceleration is dependent on the displacement, meaning that as the displacement increases, the acceleration decreases. This type of relationship is often seen in systems with a restoring force, such as a spring-mass system.

Next, we need to consider the initial conditions of the system. We are given an initial position x0, which implies that the object is at rest at this position. This means that the initial velocity of the object is zero.

Now, to calculate the displacement at time t, we can use the basic kinematic equation: x(t) = x0 + v0t + 1/2at^2. However, in this scenario, we need to take into account the fact that the acceleration is dependent on the displacement. This means that the acceleration is constantly changing as the object moves.

One way to approach this problem is to break it down into smaller time intervals. We can divide the total time t into smaller intervals, and at each interval, we can calculate the average acceleration by taking the midpoint displacement between the initial position x0 and the final position x(t). This will give us a more accurate representation of the acceleration at that specific interval.

Once we have the average acceleration for each interval, we can use the kinematic equation to calculate the displacement for each interval and then add them up to get the total displacement at time t.

Alternatively, we can use numerical methods, such as Euler's method or Runge-Kutta method, to approximate the displacement at time t. These methods involve breaking down the time interval into smaller steps and using iterative calculations to approximate the displacement at each step.

In conclusion, calculating displacement when acceleration is dependent on displacement can be a complex problem. It requires breaking down the problem into smaller intervals and using either kinematic equations or numerical methods to approximate the displacement at a specific time. As a scientist, it is important to carefully consider the initial conditions and the relationship between variables in order to approach such problems accurately.
 

1. How do you calculate displacement when acceleration is dependent on displacement?

The formula for calculating displacement when acceleration is dependent on displacement is:
d = d0 + v0t + 1/2at2
Where d is the displacement, d0 is the initial displacement, v0 is the initial velocity, a is the acceleration, and t is the time elapsed.

2. What is the difference between displacement and distance?

Displacement is a vector quantity that refers to the change in position or location of an object, while distance is a scalar quantity that refers to the total length of the path traveled by an object. Displacement takes into account the direction of motion, while distance does not.

3. How does acceleration being dependent on displacement affect an object's motion?

When acceleration is dependent on displacement, the object's motion will not be constant. Instead, the acceleration will change as the displacement changes, resulting in a non-uniform motion.

4. Can you give an example of a real-life situation where acceleration is dependent on displacement?

An example of a real-life situation where acceleration is dependent on displacement is a pendulum. As the pendulum swings back and forth, its acceleration changes depending on its displacement from the equilibrium position.

5. How does the value of acceleration affect the displacement when it is dependent on displacement?

The value of acceleration affects the displacement when it is dependent on displacement by determining how quickly the displacement changes. A higher acceleration will result in a larger change in displacement in a shorter amount of time, while a lower acceleration will result in a smaller change in displacement over a longer period of time.

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