Finding Approximate Location of Particle at Time t=1.05

• Kokordilos
In summary, The question discusses a particle's location at time t=1 and asks for its approximate location at t=1.05 using a given velocity field. The conversation then goes on to discuss the approach to solving this problem.
Kokordilos

Homework Statement

This isn't really even a homework question..i've just been reviewing some general math concepts and this one has been driving me nuts..i haven't done math in a while.

At time t = 1, a particle is located at position (1,3). If it moves in a velocity field F(x,y) = <xy-2,y^2 - 10>, find its approximate location at time t = 1.05

n/a

The Attempt at a Solution

I thought, maybe:
dx/dt = xy - 2 and dy/dt = y^2 - 10
So I can integrate, so x(t) = (1/2)x^2 * y - 2x + C and y(t) = (1/3)y^3 - 10Y + C and solving for the constants is trivial based on the initial conditions, but then I get totally confused. How do I put this in terms of T? And did I even do this right? It doesn't make sense to me that the position should be the function of a position. I must have done my integral wrong.

Can you guys help me out...really stuck.

anyone?

Here's how I would approach it, and if anyone disagrees, I hope they will jump in and correct my reasoning.

You have the velocity field F(x, y) = (xy - 2, y^2 - 10). I agree that the components are dx/dt and dy/dt.

For small values of $\Delta t$, F(x, y) $\Delta t$ should give the approximate changes in x and y.

IOW, (dx/dt, dy/dt) $\Delta t \approx (\Delta x, \Delta y)$
or (x, y) + (dx/dt, dy/dt)*dt $\approx (x, y) + (\Delta x, \Delta y)$, where x, y, dx/dt, and dy/dt are evaluated at t = 1.

Because the time increment is relatively small, the changes in x and y probably will be relatively small as well, so you should end up at a point not far from (1, 3).

1. What is the purpose of finding the approximate location of a particle at time t=1.05?

The purpose of finding the approximate location of a particle at time t=1.05 is to understand the movement and behavior of the particle at a specific point in time. This information can help in predicting future movements and studying the overall motion of the particle.

2. How is the approximate location of a particle at time t=1.05 calculated?

The approximate location of a particle at time t=1.05 is calculated by using the particle's initial position, velocity, and acceleration. These values are plugged into the equations of motion to calculate the position at the given time.

3. Can the approximate location of a particle at time t=1.05 be 100% accurate?

No, the approximate location of a particle at time t=1.05 cannot be 100% accurate. This is because it is based on a mathematical calculation and does not take into account external factors such as air resistance or friction, which can affect the actual position of the particle.

4. What are some factors that can affect the accuracy of the approximate location at time t=1.05?

The accuracy of the approximate location at time t=1.05 can be affected by factors such as measurement errors, uncertainties in initial values, and external forces acting on the particle. Additionally, the more complex the motion of the particle, the less accurate the approximation may be.

5. Is finding the approximate location of a particle at time t=1.05 useful in real-life applications?

Yes, finding the approximate location of a particle at time t=1.05 is useful in real-life applications such as studying the motion of objects in physics, predicting the trajectory of projectiles, and tracking the movement of particles in chemical reactions. It can also be used in navigation and mapping systems to determine the location of vehicles or objects at a specific time.

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