Question

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The position function of a particle at a time t is R(t)=(2 sin(π t/2(t+1)), 2 sin(π/2(t+1))). Find the velocity of the particle

Answer 1

To find the velocity of the particle at any given time, we need to take the derivative of the position function with respect to time. The velocity vector of the particle is (2π cos(π t/2(t+1)), π cos(π/2(t+1))).

Step-by-step explanation:

The position function of a particle at time t is given by R(t) = (2 sin(π t/2(t+1)), 2 sin(π/2(t+1))). To find the velocity of the particle at any given time, we need to take the derivative of the position function with respect to time. Let's start by finding the derivative of the x-component of the position function:

R'(t) = d/dt [2 sin(π t/2(t+1))] = 2π cos(π t/2(t+1))

Similarly, for the y-component:

R'(t) = d/dt [2 sin(π/2(t+1))] = π cos(π/2(t+1))

Therefore, the velocity vector of the particle is given by V(t) = (2π cos(π t/2(t+1)), π cos(π/2(t+1))).