Question

A golf ball is hit with an initial angle of \(35.5^{\circ}\) with respect to the horizontal and an initial velocity of \(83.3 \mathrm{mph} .\) It lands a distance of \(86.8 \mathrm{~m}\) away from where it was hit. By how much did the effects of wind resistance, spin, and so forth reduce the range of the golf ball from the ideal value?

Short answer

Answer: The range reduction of the golf ball due to factors such as wind resistance and spin is approximately 45.18 meters.

Step-by-step solution

Convert initial velocity to m/s

The initial velocity is given in miles per hour (mph), but we need to convert it to meters per second (m/s) for our calculations since the distance is given in meters (m). We can use the conversion: 1 mph = 0.44704 m/s. So, the initial velocity in m/s is: \(V_0 = 83.3 \mathrm{mph} \times 0.44704 \frac{\mathrm{m}}{\mathrm{s} \cdot \mathrm{mph}} \approx 37.22 \mathrm{m/s}\)

Find the horizontal and vertical components of the initial velocity

To determine the range, we need to find the horizontal and vertical components of the initial velocity. We can use the given angle and trigonometric functions to do this. \(V_{0x} = V_0 \cdot \cos{35.5^\circ} \approx 30.34 \mathrm{m/s}\) \(V_{0y} = V_0 \cdot \sin{35.5^\circ} \approx 21.34 \mathrm{m/s}\)

Find the time of flight

First, we need to determine the time it takes for the golf ball to travel vertically when there is no wind resistance. We can do this using the vertical motion equation and considering upward motion as positive: \(V_y = V_{0y} - gt\) Where \(V_y\) is the vertical velocity at the peak, \(g\) is the acceleration due to gravity (approximately \(9.81 \mathrm{m/s^2}\)), and \(t\) is the time of flight up to the peak. Since the vertical velocity at the peak is 0, we have: \(0 = 21.34 - 9.81t\) Solving for \(t\), we get the time of flight up to the peak (half the total time of flight): \(t = \frac{21.34}{9.81} \approx 2.175 \mathrm{s}\) The total time of flight is double this value: \(T = 2 \cdot t \approx 4.35 \mathrm{s}\)

Calculate the ideal range

To find the ideal range without any external influences such as wind resistance, we can multiply the horizontal component of the initial velocity by the total time of flight: \(Ideal \; Range = V_{0x} \cdot T \approx 30.34 \mathrm{m/s} \cdot 4.35 \mathrm{s} \approx 131.98 \mathrm{m}\)

Calculate the range reduction

The actual range is given as \(86.8 \mathrm{m}\). To determine the range reduction due to wind resistance, spin, and other factors, we subtract the actual range from the ideal range: \(Range \; Reduction = Ideal \; Range - Actual \; Range \approx 131.98 \mathrm{m} - 86.8 \mathrm{m} \approx 45.18 \mathrm{m}\) Therefore, the effects of wind resistance, spin, and other factors reduced the range of the golf ball by approximately \(45.18 \mathrm{m}\) from its ideal value.

Key concepts

Initial Velocity Conversion

Understanding the initial velocity of a projectile is crucial for accurate calculations in physics, especially when the given unit differs from the standard unit used in the formula. The initial velocity for projectile motion problems is often provided in miles per hour (mph), but calculations require the velocity to be in meters per second (m/s). This particular conversion uses the equivalence that 1 mph is approximately 0.44704 m/s. To convert, simply multiply the given velocity by this conversion factor.

Let's apply this to a practical situation: a golf ball hit at an initial velocity of 83.3 mph. Using the conversion factor, we calculate the initial velocity in m/s as follows:
\[\begin{equation}V_0 = 83.3 \times 0.44704 \frac{m}{s} \approx 37.22 \text{m/s}\end{equation}\]By ensuring the units are consistent, we establish a solid foundation for any subsequent calculations in projectile motion problems.

Trigonometric Functions in Physics

In projectile motion, the initial velocity is often not aligned with the standard coordinate axes. To work with its horizontal and vertical components separately, we use trigonometric functions: sine and cosine. These mathematical tools are indispensable for decomposing the initial velocity vector into its x (horizontal) and y (vertical) components.

For example, if a golf ball is hit at an angle of 35.5 degrees to the horizontal with an initial velocity (\(V_0\)), we calculate the components as:

• Horizontal component (\(V_{0x}\)): \[\begin{equation}V_{0x} = V_0 \cdot \cos(35.5^\circ)\end{equation}\]
• Vertical component (\(V_{0y}\)): \[\begin{equation}V_{0y} = V_0 \cdot \sin(35.5^\circ)\end{equation}\]

Mastering these trigonometric functions is essential for anyone studying projectile motion in physics.

Time of Flight Calculation

The time of flight refers to how long a projectile remains in the air. In an ideal scenario (ignoring air resistance), for a given initial velocity and launch angle, we can predict exactly how long a projectile will take to return to the same vertical position it was launched from. A key part of this calculation involves understanding the symmetry of projectile motion—specifically, that the time ascending to the peak is equal to the time descending from the peak.

The time of flight is twice the time it takes to reach the highest point. Assuming upward motion is positive and acceleration due to gravity (\(g\)) is about \(9.81 \text{m/s}^2\), we use the following relationship where the final vertical velocity (\(V_y\)) at the peak is zero:\[\begin{equation}0 = V_{0y} - g \times t_{\text{peak}}\end{equation}\]We then deduce the time to peak (\(t_{\text{peak}}\)) and calculate the total time of flight (\(T\)) as double this value. Understanding this concept enables one to solve many time-related problems in projectile motion.

Range Reduction in Projectile Motion

The range of a projectile is the horizontal distance it travels before landing. Calculating the ideal range assumes no external forces, like air resistance or wind, affect the projectile. However, in the real world, such factors do significantly impact the projectile's path, often causing a reduction in the range.

To determine this range reduction, we first calculate the ideal range by multiplying the horizontal component of initial velocity by the total time of flight. Then, by comparing this ideal range to the actual distance covered, measured in a real-world scenario, we can quantify the effects of external factors. For instance, if the ideal range of a golf ball is calculated to be 131.98 m but the actual range is only 86.8 m, the difference is attributed to influences like wind resistance or spin:\[\begin{equation}Range \; Reduction = Ideal \; Range - Actual \; Range\end{equation}\]This reduction of approximately 45.18 m helps us evaluate environmental or equipment factors that may need to be adjusted for better performance in sports or validated in physics experiments.