Problem 57 - Algebra 2

Question

For Exercises 56 and \(57,\) use the following information. A ball that is hit or thrown horizontally with a velocity of \(v\) meters per second will travel a distance of \(d\) meters before hitting the ground, where \(d=v \sqrt{\frac{h}{4.9}}\) and \(h\) is the height in meters from which the ball is hit or thrown. How far will a ball that is hit with a velocity of 45 meters per second at a height of 0.8 meter above the ground travel before hitting the ground?

Step-by-Step Solution

Answer

The ball will travel approximately 18.19 meters before hitting the ground.

Step 1: Identify Given Values

In this exercise, we are given a velocity \( v = 45 \) m/s and a height \( h = 0.8 \) m. We need to determine the distance \( d \) that the ball will travel before hitting the ground.

Step 2: Write Down the Formula

The formula given in the problem is \( d = v \sqrt{\frac{h}{4.9}} \). This formula allows us to calculate the distance \( d \) based on the given velocity \( v \) and height \( h \).

Step 3: Substitute Given Values Into the Formula

Replace \( v \) with 45 m/s and \( h \) with 0.8 m in the formula: \ \[ d = 45 \sqrt{\frac{0.8}{4.9}} \]

Step 4: Calculate the Square Root

Calculate the value inside the square root: \ \[ \frac{0.8}{4.9} \approx 0.1633 \] Then, find the square root: \[ \sqrt{0.1633} \approx 0.4041 \]

Step 5: Calculate the Final Distance

Now, multiply the result from the square root by the velocity \( v \): \ \[ d = 45 \times 0.4041 \approx 18.185 \] m

Step 6: Round the Distance

Round the distance to two decimal places, if necessary. So, \[ d \approx 18.19 \] m.

Key Concepts

Horizontal Velocity

Horizontal velocity is an important component in projectile motion, especially when dealing with objects launched parallel to the ground. In this context, the horizontal velocity refers to the speed at which an object travels in a horizontal trajectory. For this kind of motion, the horizontal velocity remains constant—as long as there are no forces like air resistance acting upon it.

• The horizontal velocity (\( v \)) is significant because it determines how far an object will travel horizontally.
• In our exercise example, the ball is hit with a horizontal velocity of 45 meters per second, meaning it moves at this speed right after being hit and continues until it hits the ground.

Understanding horizontal velocity helps predict the behavior of projectiles, which is crucial in many real-world applications like sports and engineering.

Distance Calculation

Distance, within projectile motion, refers to how far the object travels before hitting the ground. Calculating this distance involves considering several factors, including horizontal velocity and height from which the object is released.
To calculate distance (\( d \)), apply the formula: \[ d = v \sqrt{\frac{h}{4.9}} \]This formula combines the horizontal velocity of the object and its initial height to provide the path length until the object lands.

• For example, substituting the values given in the exercise (\( v = 45 \) m/s and \( h = 0.8 \) m) into the formula gives \[ d = 45 \sqrt{\frac{0.8}{4.9}} \]
• The calculated result, approximately 18.19 meters, shows how far the ball would travel before hitting the ground.

Performing distance calculations in physics allows for practical predictions in various fields.

Physics Formula

Physics formulas serve as tools that help convert physical concepts into mathematical expressions, allowing for precise predictions and analyses. In projectile motion, the formula for distance (\( d = v \sqrt{\frac{h}{4.9}} \)) is a key example.

Breaking Down the Formula

• The component \( v \) represents horizontal velocity, determining the lateral progress.
• The term \( \sqrt{\frac{h}{4.9}} \) accounts for the time taken, derived from gravitational influence on the vertical drop.

This formula simplifies the complex interaction between vertical and horizontal dynamics by using gravity, consistent at about \( 9.8 \text{ m/s}^2 \), in its calculation. Using such physics formulas enables precise solutions and understanding of real-world dynamics.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world scenarios, enabling the prediction of outcomes based on initial conditions. In the context of projectile motion, a mathematical model translates the mechanics of motion into calculable predictions.

Creating Effective Models

• Mathematical modeling translates physical relations like those between velocity, distance, and time into solvable equations.
• For instance, the given model \( d = v \sqrt{\frac{h}{4.9}} \) in the exercise relates to how variables interact under physics laws.

By creating models, scientists and engineers can analyze potential scenarios, guide experimentation, and innovate solutions across countless applications.