Question

A running model A model for the startup of a runner in a short race results in the velocity function \(v(t)=a\left(1-e^{-t / c}\right),\) where \(a\) and \(c\) are positive constants, \(t\) is measured in seconds, and \(v\) has units of m/s. (Source: Joe Keller, A Theory of Competitive Running, Physics Today, \(26,\) Sep 1973 ) a. Graph the velocity function for \(a=12\) and \(c=2 .\) What is the runner's maximum velocity? b. Using the velocity in part (a) and assuming \(s(0)=0\), find the position function \(s(t),\) for \(t \geq 0\) c. Graph the position function and estimate the time required to run \(100 \mathrm{m}\)

Short answer

Answer: The estimated time for the runner to complete 100 meters is approximately 12.2 seconds.

Step-by-step solution

Write the given velocity function with the given values of a and c

The velocity function is given by \(v(t) = a(1-e^{-t/c})\). We are given that \(a = 12\) and \(c = 2\). So, the velocity function becomes \(v(t) = 12(1-e^{-t/2})\).

Find out the maximum velocity

In the given velocity function, the maximum velocity occurs when the function reaches its highest value, i.e., when \(1-e^{-t/2}=1\). So, the maximum velocity will be \(v_\text{max} = 12(1-0) = 12 \mathrm{m/s}\).

Graph the velocity function

Graphing the function requires a graphing tool or software. However, you can plot the given function using any graphing tool. The function will start at \(0\mathrm{m/s}\) (when \(t=0\)) and approach \(12\mathrm{m/s}\) as \(t\) increases. #b. Finding the position function#

Integrate the velocity function

To find the position function \(s(t)\), we need to integrate the velocity function \(v(t) = 12(1-e^{-t/2})\) with respect to \(t\). We obtain: $$s(t) = \int 12(1-e^{-t/2}) dt$$

{Step 5: Calculate the integral

Calculate the integral: $$s(t) = 12 \int (1-e^{-t/2}) dt = 12\left(\int dt - \int e^{-t/2} dt\right)$$ Using integration by substitution, let \(u = -\frac{t}{2},\) so \(du = -\frac{1}{2} dt.\) Thus, we have: $$s(t) = 12\left(t - 2\int e^u (-2 du)\right) = 12\left(t + 4\int e^u du\right) = 12\left(t+4e^u\right) + C$$ Now, substitute back \(u = -\frac{t}{2}\): $$s(t) = 12\left(t + 4e^{-t/2}\right) + C$$

Determine the integration constant

We are given that \(s(0) = 0\). So, we have: $$0 = 12\left(0 + 4e^{-0/2}\right) + C \Rightarrow C = -48$$ The position function is: $$s(t) = 12\left(t + 4e^{-t/2}\right) - 48$$ #c. Graph the position function and estimate the time for 100 meters#

Graph the position function

Graphing the position function again requires a graphing tool or software. You can plot the function \(s(t) = 12\left(t + 4e^{-t/2}\right) - 48\) using any graphing tool. The function will start at \(0\mathrm{m}\) (when \(t=0\)) and will increase as \(t\) increases.

Estimate the time required to run 100 meters

To estimate the time required to run 100 meters, we need to find \(t\) when \(s(t) = 100\): $$100 = 12\left(t + 4e^{-t/2}\right) - 48$$ This equation doesn't have a simple algebraic solution, so we need to use numerical methods or a graph to find the approximate value of \(t\). Using a graphing tool or a numerical solver, you should find that \(t\) is approximately \(12.2\) seconds. Thus, the estimated time to run 100 meters is roughly \(12.2\) seconds.

Key concepts

Integrating Velocity Function

The process of integrating the velocity function is crucial for understanding how the position of an object changes over time. It involves finding the integral of the velocity with respect to time, which gives us the position function. Let’s dive deeper into the given model where the velocity function is represented by v(t) = a(1 - e^{-t/c}). To integrate this function, you follow standard calculus procedures.

For instance, for parameters a = 12 and c = 2, the specific function v(t) = 12(1 - e^{-t/2}) is integrated by treating each term separately. This involves straightforward integration for the constant part, and for the exponential part, a substitution method is typically used, as shown in the solution. After finding the indefinite integral, we determine the constant of integration using the initial conditions, which, in our case, is the position at time t = 0. This approach builds the foundation for finding the cumulative distance traveled, also known as the position function.

Graphing Position Function

Once we have the position function derived from the velocity function, graphing it can provide visual insights into the motion of the runner. For our model, the position function turned out to be s(t) = 12(t + 4e^{-t/2}) - 48 after integrating. When graphing this function, you can use graphing software to visualize how the position changes over time.

The graph typically starts with the initial position and shows a curve that represents the position increasing as time passes. The shape of the curve is important as it reflects the accelerations and decelerations of the runner. In educational settings, interpreting these graphs can help students visualize concepts like starting velocity, acceleration, and the effect of exponential decay in velocity over time—vital for a comprehensive understanding of kinematics.

Maximum Velocity Calculation

The concept of maximum velocity in the context of our velocity function model is the highest speed the runner will reach, which occurs as time goes to infinity due to the properties of the exponential function. Calculating maximum velocity is straightforward once you understand that it is the limit of the velocity function as t approaches infinity.

In the provided solution, the formula simplifies to the product of the constant a and the limit of (1 - e^{-t/c}) as t approaches infinity, which is 1. Therefore, for our given function with a = 12, the maximum velocity is 12 m/s. This value is important in physics and engineering as it represents the theoretical top speed under the given conditions and can be used to predict future motion.

Estimating Runtime in Calculus

Estimating runtime involves finding the time at which a certain event occurs, like reaching a specific position. In the context of our running model, it's about calculating when the runner will cover a certain distance, say 100 meters. This involves solving for t in the position function equation when s(t) is set to 100 meters.

Since the resultant equation from our position function doesn't lend itself to simple algebraic solutions, numerical methods or graphing tools come into play. These methods provide an approximate value of t, which in our example is around 12.2 seconds. It's a more real-world application of calculus as it requires the use of technology to handle complex expressions, teaching students the importance of computational tools in solving real-world problems.