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Chapter 13: Problem 11

A golf ball leaves the ground at a \(30^{\circ}\) angle at a speed of \(90 \mathrm{ft} / \mathrm{sec} .\) Will it clear the top of a \(30-\mathrm{ft}\) tree that is in the way, 135 \(\mathrm{ft}\) down the fairway? Explain.

Short Answer

Expert verified
Yes, the ball clears the tree; it reaches approximately 78.42 ft at the tree.

Step by step solution

01

Break Down Initial Velocity into Components

To find out if the golf ball will clear the tree, start by determining the horizontal and vertical components of the initial velocity.The formulae are: \[v_{x} = v imes ext{cos}( heta)\]\[v_{y} = v imes ext{sin}( heta)\]where \(v = 90\mathrm{ft/sec}\) and \(\theta = 30^{\circ}\). Calculating these gives:\[v_{x} = 90 imes ext{cos}(30^{\circ}) = 90 imes \frac{\sqrt{3}}{2} \approx 77.94 \mathrm{ft/sec}\]\[v_{y} = 90 imes ext{sin}(30^{\circ}) = 90 imes \frac{1}{2} = 45 \mathrm{ft/sec}\]
02

Calculate the Time to Reach the Tree

To find the time it takes for the golf ball to reach the position of the tree, use the formula:\[t = \frac{d}{v_{x}}\]where \(d = 135 \mathrm{ft}\). Plug in the values:\[t = \frac{135}{77.94} \approx 1.73 \mathrm{sec}\]
03

Determine the Height of the Ball at the Tree

Now, calculate the vertical position of the ball at \(t = 1.73\mathrm{sec}\) using the formula for vertical motion:\[y = v_{y}t - \frac{1}{2}gt^{2}\]where \(g = 32.2\mathrm{ft/sec^2}\) (acceleration due to gravity). Substituting the known values gives:\[y = 45 \times 1.73 - \frac{1}{2} \times 32.2 \times (1.73)^2 \approx 78.42\mathrm{ft}\]
04

Assess if the Ball Clears the Tree

Compare the height of the ball at the tree's position to the height of the tree. The ball reaches approximately \(78.42\mathrm{ft}\) at \(135\mathrm{ft}\) down the fairway, which is higher than the \(30\mathrm{ft}\) tree. Therefore, the ball will clear the tree.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Velocity Components
When dealing with projectile motion, the initial velocity is not directed entirely horizontally or vertically. It's split into two essential components: horizontal and vertical. This is foundational when catching a projectile like a golf ball flying through the air.
  • The **horizontal component** of velocity, represented by \(v_x\), is calculated using the cosine of the launch angle. For the golf ball, this means \(v_x = 90 \times \cos(30^{\circ})\) which simplifies to approximately **77.94 ft/sec**.
  • The **vertical component**, \(v_y\), is determined using the sine of the launch angle. So, \(v_y = 90 \times \sin(30^{\circ})\) equals **45 ft/sec**.
Understanding these components allows us to analyze how far and how high the golf ball will travel.
Horizontal Motion
Horizontal motion refers to how the projectile moves across the ground.
In our problem, the ball's horizontal motion is crucial to determine the time it takes to reach the tree.
Projectile motion assumes no air resistance, meaning the horizontal velocity component remains constant.
Using the formula \(t = \frac{d}{v_x}\), you substitute the ball's horizontal distance \(d = 135 \ \text{ft}\) and its horizontal velocity \(v_x = 77.94 \ \text{ft/sec}\).
Calculating this gives \(t \approx 1.73 \ \text{seconds}\), showing it takes approximately 1.73 seconds for the golf ball to travel horizontally to the tree.
Vertical Motion
Vertical motion involves analyzing the change in vertical position of the projectile over time. Unlike horizontal motion, vertical motion is affected by the acceleration due to gravity.
To find how high the golf ball will be when it reaches the tree, we use the formula for vertical motion: \[y = v_y t - \frac{1}{2} g t^2\] Substituting in \(v_y = 45\ \text{ft/sec}\), \(t = 1.73\ \text{sec}\), and \(g = 32.2\ \text{ft/sec}^2\), you calculate \(y \approx 78.42\ \text{ft}\).
This height shows that the ball is well above the 30-foot tree. The key aspect here is how gravity affects vertical motion, pulling the object down over time, despite its initial upward motion.
Acceleration Due to Gravity
Acceleration due to gravity is a constant \(g\), approximately **32.2 ft/sec虏** on Earth. It's a crucial factor in vertical motion, impacting how the golf ball travels upwards and then descends.
Gravity doesn't affect horizontal motion but plays a significant role in the trajectory's arch shape.Here's why it's important:
  • It causes the vertical velocity to decrease until the ball reaches its peak height and then increases as the ball falls back down.
  • Gravity is responsible for calculating how much the object is pulled downwards over time, which is part of why we subtracted \(\frac{1}{2} g t^2\) in our vertical motion formula.
Knowing the effect of gravity helps us predict and calculate the exact motion path of any projectile.

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