A particle moves on the x-axis with an acceleration, 246msta. Find the position and velocity of the particle at 3t, if the particle is at origin and has a velocity of 10ms when 0t by using either the method of undetermined coefficient or Laplace Transform.
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To solve this problem using the method of undetermined coefficients or Laplace Transform, we need to find the position \(x(t)\) and velocity \(v(t)\) of the particle given the acceleration \(a(t)\) and initial conditions.
Using the method of undetermined coefficients:
1. Start by integrating the acceleration \(a(t)\) to find the velocity \(v(t)\):
\[v(t) = \int a(t) dt = \int (246-6t) dt = 246t - 3t^2 + C_1\]
where \(C_1\) is the constant of integration.
2. Now, integrate the velocity \(v(t)\) to find the position \(x(t)\):
\[x(t) = \int v(t) dt = \int (246t - 3t^2 + C_1) dt = 123t^2 - t^3 + C_1t + C_2\]
where \(C_2\) is the constant of integration.
3. To determine the values of \(C_1\) and \(C_2\), we use the initial conditions:
If the particle is at the origin (\(x(0) = 0\)), we have \(C_2 = 0\).
If the particle has a velocity of \(-10 \, \text{m/s}\) at \(t = 0\) (\(v(0) = -10 \, \text{m/s}\)), we have:
\[-10 = 0 - 0 + C_1 \Rightarrow C_1 = -10\]
Therefore, the position \(x(t)\) and velocity \(v(t)\) of the particle at \(t = 3\) can be found by substituting \(t = 3\) into the equations found in step 2 and step 1, respectively:
\[v(3) = 246(3) - 3(3)^2 - 10\]
\[x(3) = 123(3)^2 - (3)^3 + (-10)(3)\]
Solving these equations will give you the values of \(v(3)\) and \(x(3)\).
Using Laplace Transform:
Alternatively, we can solve this problem using Laplace Transform, which involves transforming the given differential equations into algebraic equations and then inversely transforming them to obtain the solution.
1. Apply the Laplace Transform to both sides of the equation \(a(t) = 246 - 6t\).
2. Solve the resulting algebraic equation for the Laplace Transform of velocity \(V(s)\).
3. Inverse transform \(V(s)\) to obtain the velocity function \(v(t)\).
4. Integrate \(v(t)\) to find the position function \(x(t)\).
Similarly, substitute \(t = 3\) into the obtained functions to find \(v(3)\) and \(x(3)\).
Both methods should yield the same solution for the position and velocity of the particle at \(t = 3\) seconds.