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Chapter 16: Problem 10

Compared to \(R\) at an initial speed of \(9 \mathrm{ft} / \mathrm{sec}, R\) at an initial speed of \(18 \mathrm{ft} / \mathrm{sec}\) is: F. \(1 / 4\) as great. G. \(1 / 2\) as great. H. 2 times as great. J. 4 times as great.

Short Answer

Expert verified
The range of the projectile launched with an initial speed of 18 ft/sec is 4 times as great as the range of the projectile launched with an initial speed of 9 ft/sec. Therefore, the correct answer is J. 4 times as great.

Step by step solution

01

Calculate range ratio between two projectiles with different initial speeds

We will use the formula for the range of a projectile: \[R = \frac{v^2 \sin(2\theta)}{g}\] First, find the range of the projectile with an initial speed of 9 ft/sec: \[R_1 = \frac{(9)^2 \sin(2\theta)}{g}\] Now, find the range of the projectile with an initial speed of 18 ft/sec: \[R_2 = \frac{(18)^2 \sin(2\theta)}{g}\] Next, we want to find the ratio between the ranges of the two projectiles: \(\frac{R_2}{R_1}\).
02

Calculate the range ratio

Divide the range of projectile 2 by the range of projectile 1 to find the ratio: \[\frac{R_2}{R_1} = \frac{\frac{(18)^2 \sin(2\theta)}{g}}{\frac{(9)^2 \sin(2\theta)}{g}}\] The \(\sin(2\theta)\) and \(g\) terms cancel out: \[\frac{R_2}{R_1} = \frac{(18)^2}{(9)^2}\] Solve the ratio: \[\frac{R_2}{R_1} = \frac{324}{81} = 4\] Therefore, the range of the projectile launched with an initial speed of 18 ft/sec is 4 times as great as the range of the projectile launched with an initial speed of 9 ft/sec.
03

Choose the correct answer

From our calculation, we found that the range of the projectile with an initial speed of 18 ft/sec is 4 times as great as the range of the projectile with an initial speed of 9 ft/sec. The correct answer is: J. 4 times as great.

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Key Concepts

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

Projectile Motion
Projectile motion refers to the motion of an object that is launched into the air and influenced only by the force of gravity and its initial thrust. It's common in both natural and artificial scenarios, such as when you throw a ball or launch a rocket.
To understand projectile motion better, consider these key points:
  • Two components: The motion can be divided into horizontal and vertical components. The horizontal component involves constant velocity, while the vertical component involves constant acceleration due to gravity.
  • Path followed: The path of a projectile is a parabola in ideal situations where we ignore air resistance, meaning it moves upwards and then comes back down following a curved trajectory.
  • Factors affecting motion: The initial velocity, launch angle, and the height from which it's projected all influence its path.
Understanding these components helps diagnose why projectile range can vary significantly with differing initial speeds, making it a fascinating subject in physics and math.
Range Formula
The range formula is crucial in determining how far a projectile travels horizontally before landing. The mathematical representation of the range formula is:\[ R = \frac{v^2 \sin(2\theta)}{g} \]
Breaking it down:
  • \(v\): The initial velocity of the projectile. It's squared in the formula, highlighting its significant impact on range.
  • \(\sin(2\theta)\): A trigonometric function that accounts for the launch angle \(\theta\). This implies that range is maximized when the angle is 45 degrees, where \(\sin(2\theta) = \sin(90) = 1\).
  • \(g\): The acceleration due to gravity, which on Earth is approximately 9.8 m/s虏 or 32 ft/s虏. It inversely affects the range, meaning higher gravity reduces range.
Using this formula, you can solve various problems regarding how initial speeds influence how far a projectile travels. Understanding it is essential for analyzing real-world scenarios involving trajectories.
Ratio Calculation
In the context of projectile motion, ratio calculation can help compare the range of different projectiles with varying initial speeds. Here, we use the ratio:
To find the relationship between two projectiles with initial speeds of 9 ft/sec and 18 ft/sec, plug into the formula:\[ \frac{R_2}{R_1} = \frac{(18)^2}{(9)^2} \]
  • Simplifying the ratio: You calculate \((18)^2 = 324\) and \((9)^2 = 81\), and then divide: \(\frac{324}{81} = 4\).
This outcome means the range of a projectile with an initial speed of 18 ft/sec is 4 times greater than that of 9 ft/sec.
Ratio calculations like these are instrumental in understanding how altering initial conditions affects projectile motion.
Initial Speed Comparison
Initial speed is a key determining factor in the behavior of projectile motion. In comparing initial speeds of two projectiles, especially in range analysis, it gives insight into how energy and speed correlate with distance traveled.
Some important considerations with initial speed include:
  • A higher initial speed, as derived from the range formula, significantly increases the projectile's range, assuming other factors remain constant.
  • Doubling the initial speed doesn't simply double the range; rather, as seen in the ratio equation, it can quadruple the range. This relation is because the initial speed is squared in the range formula: \(v^2\).
  • Careful adjustments to initial speed can result in more efficient outcomes in practical applications, such as sports or engineering.
Understanding initial speed comparison is vital for predicting outcomes in projectile motion scenarios, enabling more precise modeling and planning.

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