The rectilinear motion of a particular is defined by the displacement time equationx = x0[2e−kt−e−2kt]Where x0 is the initial displacement, K is a constant. What will be the maximum velocity of the particle?
Question
The rectilinear motion of a particular is defined by the displacement time equationx = x0[2e−kt−e−2kt]Where x0 is the initial displacement, K is a constant. What will be the maximum velocity of the particle?
Solution
To find the maximum velocity of the particle, we first need to find the velocity function. The velocity of the particle is the derivative of the displacement function with respect to time.
The displacement function is given as x = x0[2e^(-kt) - e^(-2kt)].
Let's differentiate this with respect to time (t) to get the velocity function (v):
v = dx/dt = x0[-2ke^(-kt) + 2ke^(-2kt)]
Now, to find the maximum velocity, we need to find the critical points of the velocity function. The critical points occur where the derivative of the velocity function is equal to zero or undefined.
Let's differentiate the velocity function with respect to time (t) to get the acceleration function (a):
a = dv/dt = x0[2k^2e^(-kt) - 4k^2e^(-2kt)]
Set a = 0 and solve for t to find the critical points:
2k^2e^(-kt) - 4k^2e^(-2kt) = 0
Solving this equation will give the values of t at which the velocity is maximum. Substitute these values of t in the velocity function to get the maximum velocity.
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