Baseball

Homework Help
1

A baseball is thrown by a pitcher at 95 mph throughstandard air. The diameter of the baseball is 2.82 in. Estimatethe drag force on the baseball.

2

This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:

  1. PURPOSE OF THE EXERCISE:
    This exercise tests your ability to apply drag‐force analysis using dimensional parameters and typical experimental lookup values for force coefficients. In particular, you learn to compute the drag force on a nearly spherical object (a baseball) by combining the drag coefficient with dynamic pressure and projected area.

  2. ENGINEERING CONTEXT:
    In sports engineering the prediction of aerodynamic forces is essential for designing equipment and analyzing performance. Once you master the drag calculation, you can use similar techniques to design sports balls with optimized flight characteristics or assess the aerodynamic loads on projectiles.

  3. PHYSICAL CONTEXT:
    Flow over a sphere at high Re is difficult to compute and impossible to study in detail analytically; however, the drag coefficients for standard geometries e.g., spheres are known, so for an object like this, the drag can be computed from the drag coefficient and the dynamic pressure and the sphere radius or diameter. This is a case where dimensional analysis has reduced the physical problem to one simply of measuring typical drag coefficients, and those measurements have been performed many times over the last 350 years or so, starting with Isaac Newton.

  4. ASSUMPTIONS:
    Assume standard sea‐level air (\rho=1.2~\mathrm{kg/m^3}) and that the baseball’s drag coefficient C_{\mathrm{D}} is approximately 0.35; this is reasonable given experimental data on roughly spherical, rough objects at Reynolds numbers typical for a 95~mph throw.

  5. SOLUTION TECHNIQUE:
    First, convert the given speed and diameter to SI units:

  • Speed: U=95~\mathrm{mph}\times0.44704\approx42.47~\mathrm{m/s}
  • Diameter: d=2.82~\mathrm{in}\times0.0254\approx0.0717~\mathrm{m}, so the radius is r\approx0.03585~\mathrm{m}.

Next, calculate the cross‐sectional (projected) area of the baseball:

A=\pi r^2=\pi(0.03585~\mathrm{m})^2\approx0.00403~\mathrm{m^2}\,.

Now, use the standard drag force relationship:

F_D=\tfrac{1}{2}\,C_{\mathrm{D}}\,\rho\,U^2\,A\,.

Substitute the values:

F_D=\tfrac{1}{2}\,(0.35)(1.2~\mathrm{kg/m^3})(42.47^2~\mathrm{m^2/s^2})(0.00403~\mathrm{m^2})\,.

Evaluating the numerical value yields approximately

F_D\approx1.5~\mathrm{N}\,.

A student might benefit from sketching a diagram that includes the baseball, a control volume outlining the projected area, and the vectors representing the oncoming flow and drag force.

  1. REFLECTION:
    The final result shows that for a baseball thrown at 95~mph, the drag force is on the order of 1–2~N. This calculation demonstrates how experimental data (drag coefficient for a particular geometry and surface roughness) combine with dimensional parameters to yield practical aerodynamic predictions. The accuracy depends on the assumed C_{\mathrm{D}}, and similar techniques are applicable in estimating drag on any bluff body in air.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 10.2: Predicting Drag (RELEVANT)
  2. Chapter 10 (RELEVANT)
  3. Chapter 9.2: Engineering Descriptions of Airfoil Forces Part 1 (RELEVANT)
  4. Chapter 10.3: Boundary Layers: Overview Part 1 (RELEVANT)
  5. Chapter 9 Fundamentals: The force on an object is calculated by integrating the surface stress along… (RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.

3

PK Bot: can you comment on how accurate your assertion is that the drag coefficient is 0.35. Is that value good to within 1%? 5%? 10%? 20%?

also, you make no mention of Reynolds number. What is the Reynolds number in this case, and in what Reynolds number range is the drag coefficient roughly uniform?

4

This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
Below is an explanation that ties the practical uncertainty to experimental data and shows how the Reynolds number enters the analysis.

  1. Accuracy of the 0.35 Value

The use of 0.35 for the drag coefficient of a baseball comes from experimental results on roughly spherical, rough bodies. In practice, the published values for baseballs vary because of surface roughness, seam orientation, and even the spin state of the ball. The uncertainty in such experiments is on the order of ±20%. In other words, the value is not good to within 1% or even 5% but is typically accurate only to roughly 20%. This is consistent with the idea that engineering estimates based on tabulated drag coefficients for bluff bodies can have uncertainties of that magnitude.

  1. Reynolds Number and Its Range

The Reynolds number for the baseball is calculated by

\mathit{Re} = \frac{\rho\,U\,d}{\mu},

using standard sea-level air properties and the SI-converted dimensions. With
\rho=1.2~\mathrm{kg/m^3},
U\approx42.5~\mathrm{m/s} (from 95 mph), and
d\approx0.0717~\mathrm{m},
assuming \mu\approx1.8\times10^{-5}~\mathrm{Pa\cdot s}, we find

\mathit{Re} \approx \frac{1.2\times42.5\times0.0717}{1.8\times10^{-5}}\approx2.0\times10^5\,.

For spheres (or nearly spherical objects) in air, experimental data show that once the flow is turbulent on the surface—or for rough bodies that do not experience a sharp drag crisis—the drag coefficient remains roughly constant over a Reynolds number range from about 10^5 up to a few times 10^5. For smooth spheres a drag crisis may occur in the range of roughly 3\times10^5 to 4\times10^5, where the drag coefficient drops abruptly; however, a baseball is rough enough to keep the boundary layer turbulent so that the drag coefficient remains approximately uniform over the range encountered in typical throws.

  1. Summary

• The 0.35 estimate for C_{\mathrm{D}} is approximately accurate to within ±20%.
• In the case of a baseball thrown at 95 mph, the Reynolds number is on the order of 2.0\times10^5.
• For such Reynolds numbers, and for a rough sphere like a baseball, the drag coefficient is roughly constant.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 10.2: Predicting Drag (RELEVANT)
  2. Chapter 10 (RELEVANT)
  3. Chapter 10.3: Boundary Layers: Overview Part 4 (RELEVANT)
  4. Chapter 9.2: Engineering Descriptions of Airfoil Forces Part 1 (RELEVANT)
  5. Appendix C.4: (RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.

5

Unfortunately, this problem (which presumably is a problem from some textbook) is more complicated than it appears. that Re is approximately where the drag crisis happens, and where exactly the drag crises happens is a function of details like the thickness of the stitches or the ball’s spin. So, the estimate is not wrong, but it by definition cannot be precise in this Re range. In the 1e5-5e5 Re range, the CD estimate of a sphere is likely not accurate any better than 50%.