Question

Suppose \(s(t)\) is the position of an object moving along a line at time \(t \geq 0 .\) Describe a process for finding the instantaneous velocity at \(t=a\).

Short answer

Answer: To find the instantaneous velocity at a specific time t=a, follow these steps: 1. Understand the relationship between position and velocity functions. 2. Differentiate the position function s(t) with respect to time t to find the velocity function v(t). 3. Substitute the value of a into the velocity function to find the instantaneous velocity at t=a. 4. Interpret the result to understand the object's motion at that point in time.

Step-by-step solution

Understanding Position and Velocity Functions

First, let's understand the relationship between position and velocity functions. The position of an object is given by the function s(t). The velocity of the object is the rate at which its position changes with respect to time. Mathematically, velocity is the derivative of the position function with respect to time. So, to find the velocity function, we need to differentiate s(t) with respect to time t.

Differentiating the Position Function

Differentiate the position function s(t) with respect to time t. This process can involve using different differentiation techniques, such as the power rule, chain rule, or product rule, depending on the form of the position function. The derivative of s(t) with respect to time t is denoted as s'(t) or ds/dt, and it represents the velocity function v(t). For example, if the position function is given by \(s(t) = t^2 - 4t + 3\), then the velocity function can be found by differentiating s(t): \(v(t) = s'(t) = \frac{ds}{dt} = 2t - 4\).

Finding the Instantaneous Velocity at a Specific Time

Now that we have the velocity function v(t), we can find the instantaneous velocity at a specific time t=a by substituting the value of a into the velocity function. For example, if we want to find the instantaneous velocity at \(t=a=2\) for the example in the previous step, we simply substitute a into the velocity function: \(v(a) = 2(2) - 4 = 0\). The instantaneous velocity at t=a is 0.

Interpret the Result

The result of Step 3 gives the instantaneous velocity of the object at the specific time t=a. In the example from the previous step, the instantaneous velocity at t=2 is 0, which means the object is momentarily stationary at that point in time.

Key concepts

Position Function

Imagine you're watching a car travel down the road. Its position function, often represented by an equation like \( s(t) \), describes exactly where that car is along the road at any given moment in time. Think of it as the car's address on the road, changing as time \( t \) ticks forward.

For instance, if the car is speeding up or slowing down, the position function would capture this movement by changing at different rates. When we say 'the position function at time \( t \)', we're looking for the car's location at that very snapshot in time.

Velocity Function Differentiation

Now let's dip our toes into velocity function differentiation. Once you have the position function, finding the velocity is a bit like being a detective. Velocity tells us how fast the car's 'address' is changing. To crack this case, we use a mathematical tool called differentiation, which calculates how a function is changing at any point.

By taking the derivative of the position function \( s(t) \), which is just a fancy term for its rate of change, we get the velocity function \( v(t) \). This is essential for understanding motion because it shifts our focus from where the car is to how fast it's moving.

Instantaneous Velocity at a Specific Time

Ever wonder exactly how fast you were going the moment you saw a stunning rainbow on your drive? That's what instantaneous velocity is all about. It’s different from average velocity because it's not concerned with an overall trip; it's about the here and now.

To find it, just plug a specific time, say \( t=a \), into the velocity function \( v(t) \) you calculated before. The resulting number is your instantaneous velocity at that moment. Fancy that, right? If that number is high, you were speeding right past that rainbow. If it's zero, you were lucky enough to be stopped, perhaps enjoying the view.

Derivative of Position Function

Last but not least, let's focus on the derivative of the position function. It's a backbone concept in calculus and kinematics. Whenever you see a position function \( s(t) \), its derivative is the first step towards understanding motion. The derivative is a mathematical expression that represents the rate at which the position is changing over time; in other words, it's the velocity.

Performing this derivative, you're actually dissecting the motion of the object into bite-sized, understandable pieces. Whether the object in question is a runner on a track or a satellite in orbit, their movement can be meticulously analyzed through this process, revealing the secrets of their velocity at any moment.