Question

Consider a particle moving along the \(x\) -axis where \(x(t)\) is the position of the particle at time \(t, x^{\prime}(t)\) is its velocity, and \(x^{\prime \prime}(t)\) is its acceleration. A particle, initially at rest, moves along the \(x\) -axis such that its acceleration at time \(t>0\) is given by \(a(t)=\cos t\). At the time \(t=0\), its position is \(x=3\). (a) Find the velocity and position functions for the particle. (b) Find the values of \(t\) for which the particle is at rest.

Short answer

The velocity function is \(v(t) = \sin(t)\), the position function is \(x(t) = -\cos(t) + 4\), and the particle is at rest for \(t = n\pi\) where \(n\) is an integer.

Step-by-step solution

Integration of Acceleration to Get Velocity

Since the velocity is the integral of the acceleration, we will perform the integral \(\int a(t) dt\). So, \(v(t) = \int\cos(t) dt = \sin(t) + C\), where \(C\) is the constant of integration. Since the initial velocity is 0, that is \(v(0) = 0\), from \(v(t)\) equation we can find that \(C = 0\). So, the velocity function is \(v(t) = \sin(t)\).

Integration of Velocity to Get Position

The position is the integral of the velocity, so \(x(t) = \int v(t) dt = \int \sin(t) dt = -\cos(t) + D\), where \(D\) is the constant of integration. We know that the initial position is \(x(0) = 3\), so we substitute \(t = 0\) into \(x(t)\) to solve for \(D = 4\). Thus, the position function is \(x(t) = -\cos(t) + 4\).

Solve When Particle is at Rest

The particle is at rest when the velocity function \(v(t) = 0\). So, we solve for \(t\) in the equation \(\sin(t) = 0\), which gives us \(t = n\pi\), where \(n\) is an integer.