Question

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=-32 ; v(0)=20, s(0)=0$$

Short answer

Answer: The position function of the object is $$s(t) = -16t^2 + 20t$$.

Step-by-step solution

Integrate acceleration function, find velocity function

Integrate the acceleration function $$a(t) = -32$$ to find the velocity function $$v(t)$$: $$v(t) = \int a(t) dt = \int (-32) dt = -32t + C$$ where C is the integration constant. We are given the initial velocity $$v(0) = 20$$. Substitute t = 0 and v(0) into the equation, to solve for C: $$20 = -32(0) + C \Rightarrow C = 20$$ The velocity function is then: $$v(t) = -32t + 20$$

Integrate velocity function, find position function

Next, we integrate the velocity function $$v(t) = -32t + 20$$ to find the position function $$s(t)$$: $$s(t) = \int v(t) dt = \int (-32t + 20) dt = -16t^2 + 20t + D$$ where D is another integration constant. We are given the initial position $$s(0) = 0$$. Substitute t = 0 and s(0) into the equation, to solve for D: $$0 = -16(0)^2 + 20(0) + D \Rightarrow D = 0$$ The position function is then: $$s(t) = -16t^2 + 20t$$

Evaluate position function

Now we have the position function for the given problem: $$s(t) = -16t^2 + 20t$$ This function describes the position of the object along the line at any given time t, taking into account the provided acceleration function, initial velocity, and initial position.

Key concepts

Integration of Acceleration

Understanding how to integrate an acceleration function is crucial in calculus to find the velocity function of a moving object. Integration is the inverse process of differentiation. If we know the acceleration, which is the rate of change of velocity, we can integrate it to find the velocity at any point in time.

When given an acceleration function such as \(a(t) = -32\), we integrate with respect to time to obtain the velocity function. Integration adds a constant of integration, labeled commonly as \(C\), because indefinite integrals do not specify limits and provide a family of solutions. To find this constant, we use known conditions such as the initial velocity, which provides a specific solution for our problem.

Initial Velocity

The initial velocity, often denoted as \(v(0)\), is the velocity of the object at time zero or the starting velocity. In the context of integrating an acceleration function, the initial velocity is used to find the integration constant, \(C\).

We integrate the acceleration \(a(t)\) to get a general form of the velocity function inclusive of \(C\). By plugging the initial velocity value into this equation, we solve for \(C\), making the equation specific to the motion of the object being described. In our example, an initial velocity of \(20\) m/s means when \(t=0\), \(v(0) = -32(0) + C = 20\), hence \(C = 20\).

Initial Position

Similarly, the initial position, indicated by \(s(0)\), is the position of the object at time zero. It is the starting point from which we measure the object's subsequent movement. After finding the velocity function, we integrate it to obtain the position function. The position function will also have an integration constant, often represented as \(D\).

To find \(D\), we substitute \(t = 0\) and the known initial position into the position function, \(s(0)\). For instance, with an initial position \(s(0) = 0\), we find that \(D = 0\). This simplifies our position function and links it specifically to the object's motion.

Velocity Function

The velocity function \(v(t)\) represents how the velocity of an object changes over time. It is derived from integrating the acceleration function and applying initial conditions to determine integration constants. The velocity function gives us insights into an object's behavior, such as when and whether it stops, reverses direction, or maintains a constant speed. In our exercise, after integration and applying the initial velocity, the resulting function \(v(t) = -32t + 20\) tells us the object’s velocity at any time \(t\) and how it changes due to the constant acceleration.