Question

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=2 \cos t ; v(0)=1, s(0)=0$$

Short answer

Answer: The position function of the object is \(s(t) = -2\cos t + t + 2\).

Step-by-step solution

Integrate the acceleration function

Integrate the acceleration function \(a(t) = 2\cos t\) with respect to time t to obtain the velocity function \(v(t)\): $$v(t) = \int a(t) dt = \int 2\cos t dt$$ The integral of \(\cos t\) is \(\sin t\), so we have: $$v(t) = 2\sin t + C_{1}$$ where \(C_{1}\) is the constant of integration.

Apply the initial velocity condition

Apply the initial velocity condition \(v(0) = 1\) to find the constant \(C_{1}\): $$1 = 2\sin(0) + C_{1}$$ Since \(\sin(0) = 0\), we have: $$C_{1} = 1$$ Thus, the velocity function becomes: $$v(t) = 2\sin t + 1$$

Integrate the velocity function

Integrate the velocity function \(v(t) = 2\sin t + 1\) with respect to time t to obtain the position function \(s(t)\): $$s(t) = \int v(t) dt = \int (2\sin t + 1) dt$$ The integral of \(\sin t\) is \(-\cos t\), and the integral of 1 is t. So we have: $$s(t) = -2\cos t + t + C_{2}$$ where \(C_{2}\) is the constant of integration.

Apply the initial position condition

Apply the initial position condition \(s(0) = 0\) to find the constant \(C_{2}\): $$0 = -2\cos(0) + 0 + C_{2}$$ Since \(\cos(0) = 1\), we have: $$C_{2} = 2$$ Thus, the position function becomes: $$s(t) = -2\cos t + t + 2$$ The position function of the object is \(s(t) = -2\cos t + t + 2\).

Key concepts

Acceleration Function

The acceleration function provides insight into how the velocity of an object changes over time. It is a derivative of the velocity function with respect to time. In the given exercise, the acceleration function is expressed as \(a(t) = 2\cos t\). This function describes the rate of change of velocity for the object moving along a line.
To find the velocity function from the acceleration function, we need to integrate the acceleration function over time. This process involves finding an antiderivative, which effectively "undoes" the differentiation that defined the acceleration from the velocity. The result of this integration gives the velocity as a function of time, including a constant of integration, since integration is the reverse of differentiation.

• In our example, the initial velocity condition \(v(0) = 1\) helps to determine this constant, ensuring that the velocity function accurately reflects the object's motion from the outset.

Velocity Function

The velocity function shows how an object’s position changes over time. It is an integral of the acceleration function. In our exercise, the velocity function is derived by integrating \(a(t) = 2\cos t\), leading to \(v(t) = 2\sin t + C_1\).
Determining the constant \(C_1\) is crucial. The initial condition \(v(0) = 1\) is given to solve for \(C_1\), meaning at time \(t = 0\), the velocity is 1. Substituting into the equation gives:
\[1 = 2\sin(0) + C_1\]Since \(\sin(0) = 0\), it simplifies to \(C_1 = 1\). Thus, the full velocity function is \(v(t) = 2\sin t + 1\). This function now accurately represents the rate of position change tailored for the given initial condition, allowing us to predict future movement of the object along the line.

Integration Techniques

Integration techniques are key to moving from acceleration to velocity, and from velocity to position. Each step involves integrating a function and applying initial conditions to find constants.
In our exercise, after deriving the velocity function \(v(t) = 2\sin t + 1\), we integrate again to get the position function. This step involves integrating both terms:

• The integral of \(2\sin t\) is \(-2\cos t\).
• The integral of a constant 1 is the linear function \(t\).

This integration results in the expression \(s(t) = -2\cos t + t + C_2\). Applying the condition \(s(0) = 0\) helps solve for \(C_2\):
\[0 = -2\cos(0) + 0 + C_2\]Given \(\cos(0) = 1\), we compute \(C_2 = 2\). Thus, the position function becomes \(s(t) = -2\cos t + t + 2\). This comprehensive approach ensures that all function transitions – from acceleration to velocity to position – adhere to the initial conditions specified in the problem.