Question

Use the position function \(s(t)=-4.9 t^{2}+150\), which gives the height (in meters) of an object that has fallen from a height of 150 meters. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). Find the velocity of the object when \(t=3\).

Short answer

The velocity of the object when \(t=3\) is -29.4 m/s

Step-by-step solution

Calculate s(a) and s(t)

Calculate \(s(a) = -4.9a^{2}+150\) and \(s(t) = -4.9t^{2}+150\). Here \(a\) will represent \(t=3\).

Develop Velocity Function

The velocity function is given by the limit as \(t\) approaches \(a\) of \(\frac{s(a)-s(t)}{a-t}\). Substitute the values from step 1 into this formula to get the function \(\lim _{t \rightarrow a} \frac{-4.9a^{2}+150 - (-4.9t^{2}+150)}{a-t}\). The terms involving 150 in the numerator will cancel out, simplifying the function to \(\lim _{t \rightarrow a} \frac{-4.9a^{2}+4.9t^{2}}{a-t}\). Factor out \(-4.9\) and reverse the order in the numerator so that the limit will exist, resulting in \(-4.9 \lim _{t \rightarrow a} \frac{t^{2}-a^{2}}{t-a}\). This limit is in the form of \(\frac{0}{0}\) when \(t = a\), hence by limit property, rewrite the limit as \(-4.9 \lim _{t \rightarrow a} \frac{(t-a)(t+a)}{t-a}\), where the term \(t-a\) in numerator and denominator will cancel each other. Thus, the velocity function will become \(-4.9 \lim _{t \rightarrow a} (t+a)\).

Find the Velocity at t=3

To find the velocity at \(t=3\), substitute \(a=3\) into the velocity function from step 2. This gives the expression \(-4.9 * (3+3)\), which simplifies to -29.4 m/s.

Key concepts

Position Function

In the world of physics and calculus, the position function is a foundational concept that tells us the location of an object at any given time. In our exercise, the position function is given as \(s(t)=-4.9t^2+150\), which describes the height, in meters, of an object as it falls from a height of 150 meters over time \(t\).

This quadratic form is a result of integrating the acceleration due to gravity, -9.8 m/s² (which is halved in the equation to -4.9 m/s² considering the initial velocity is zero for a free-falling object), with respect to time to obtain the velocity and once more to get the position. Understanding how to manipulate this function allows us to analyze the motion of the object, like calculating its velocity at a certain time which is a crucial concept in both physics and calculus.

Instantaneous Velocity

Instantaneous velocity is a measure of how fast an object is moving at a specific instant in time. Unlike average velocity, which is the total displacement divided by total time, instantaneous velocity focuses on a single moment. The formula provided in the original exercise, \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\), embodies the concept of instantaneous velocity as the limit of the average velocity as the time interval approaches zero.

In simpler terms, it's like zooming in on a stopwatch very closely to see the speed of an object at an exact second. When we calculate this limit, we uncover the object's speed and direction (indicated by the sign of the value) at the specific time \(t=a\). This task is akin to finding out how fast your car was going the moment you looked at the speedometer, not your average speed during the entire trip.

Limit of a Function

The limit of a function is a fundamental notion in calculus, crucial for understanding instantaneous rates of change like velocity. It describes the value that a function approaches as the input (or 'x' value) approaches a certain point. Limits can provide answers even when direct substitution isn’t possible due to indeterminate forms, such as \(\frac{0}{0}\).

In the context of our exercise, by setting up the limit \(\lim _{t \rightarrow a} \frac{t^{2}-a^{2}}{t-a}\), we deal with an indeterminate form that needs simplification. We apply algebraic manipulation, factoring out a difference of squares, to cancel out the troublesome parts and then evaluate the limit. The tools of limits allow us to smoothly transition from an average velocity over a 'small' time interval to the instantaneous velocity at a precise moment, making it the bridge between average rates and instantaneous measurements in motion analysis.