Question

A golfer gives a ball a maximum initial speed of \(34.4 \mathrm{~m} / \mathrm{s}\). (a) What is the longest possible hole-in-one shot for this golfer? Neglect any distance the ball might roll on the green and assume that the tee and the green are at the same level. (b) What is the minimum speed of the ball during this hole-in-one shot?

Short answer

(a) The longest hole-in-one shot is approximately 120.8 meters. (b) The minimum speed of the ball is approximately 24.3 m/s.

Step-by-step solution

Determine the Ideal Launch Angle

To achieve the longest possible shot distance, the launch angle should be at 45 degrees. This angle maximizes the horizontal range of a projectile in uniform motion.

Use the Range Formula for Projectiles

The range of a projectile (R) is given by the formula:\[ R = \frac{v^2 \sin(2\theta)}{g} \]where \( v = 34.4 \text{ m/s} \), \( \theta = 45^\circ \), and \( g = 9.81 \text{ m/s}^2 \).Substitute these values into the formula:\[ R = \frac{(34.4)^2 \sin(90^\circ)}{9.81} \]Since \( \sin(90^\circ) = 1 \), we get:\[ R = \frac{(34.4)^2}{9.81} \approx 120.8 \text{ meters} \]

Determine the Minimum Speed of the Ball

The minimum speed of the ball occurs at the highest point of its trajectory. At this point, the vertical component of the velocity is zero, and only the horizontal component remains. The horizontal component \( v_x \) is given by:\[ v_x = v \cos(\theta) \]Substituting the values, we get:\[ v_x = 34.4 \cos(45^\circ) = 34.4 \times \frac{1}{\sqrt{2}} \approx 24.3 \text{ m/s} \]

Key concepts

Launch Angle

The launch angle plays a critical role in determining how far a projectile will travel, especially in cases such as a golfer hitting a ball. When discussing the optimal launch angle for maximum distance, a specific value comes into play. For a projectile launched in uniform motion, a 45-degree angle is often ideal. This angle allows the horizontal and vertical velocity components to be perfectly balanced, ensuring that the projectile travels the furthest possible distance.

To better understand this, consider that at a 45-degree angle:

• The sin component, used in calculating range, reaches its maximum possible value.
• The trajectory path is symmetrical, maximizing horizontal distance.
• The energy is optimally divided between height and distance.

Hence, mastering the concept of launch angle is essential for tasks like determining the longest possible golf shot.

Projectile Range

The term projectile range refers to the horizontal distance a projectile covers during its flight. In our golf example, finding the range is crucial to determine how far the ball will travel before landing. The range for a projectile can be calculated using the formula:

• \[ R = \frac{v^2 \sin(2\theta)}{g} \]

Where:

• \( v \) is the initial speed of the projectile.
• \( \theta \) is the launch angle.
• \( g \) is the acceleration due to gravity.

Plugging in the values from the exercise, we achieve the longest possible shot calculation, resulting in approximately 120.8 meters.

This range assumes zero air resistance and level ground, as simplifications facilitate an analysis of projectile motion. Understanding and using this formula can be beneficial in many applications, not just golf.

Velocity Components

Understanding the velocity components of a projectile is crucial in determining its motion characteristics. A projectile's initial velocity can be divided into two components: horizontal and vertical. When a ball is hit, these components influence its trajectory:

• Horizontal Velocity Component (\( v_x \)): This remains constant during flight as it is unaffected by gravity. It is calculated as: \[ v_x = v \cos(\theta) \]
• Vertical Velocity Component ( \( v_y \)): This changes due to gravity, affecting the projectile until its ascent and descent phase switches. Initially, it's found by: \[ v_y = v \sin(\theta) \]

In our exercise, at the maximum height, the vertical component is zero, so only the horizontal component remains, being approximately 24.3 m/s.

Recognizing and calculating these components are key in predicting how any projectile, such as a golf ball, will travel.

Maximum Range

Maximum range is achieved when a projectile attains the greatest possible horizontal distance. In the exercise scenario, this means the longest hole-in-one shot the golfer can achieve. To attain this, certain conditions are necessary:

• The launch angle must be 45 degrees, which is optimal for maximum range balancing between height and distance.
• The projectile's initial velocity must be used fully to maximize the distance.
• No additional forces like air resistance impact the projectile.

By using these principles, we calculated the maximum range the golfer can achieve, which is approximately 120.8 meters. The knowledge of maximum range is not only beneficial in golfing but also in other physics problems involving projectiles.