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If a baseball pitcher throws a fastball at a horizontal speed of \(160 \mathrm{~km} / \mathrm{h},\) how long does the ball take to reach home plate \(18.4 \mathrm{~m}\) away?

Short Answer

Expert verified
The ball takes approximately 0.414 seconds to reach home plate.

Step by step solution

01

Convert Speed to Meters per Second

First, we need to convert the speed from kilometers per hour to meters per second, since we are dealing with meters for distance. The conversion factor is 1 km/h = 0.27778 m/s.Calculate the speed: \[ 160 \times 0.27778 = 44.4448 \text{ m/s} \]
02

Calculate Time Using the formula

Use the formula for time, which is:\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]Here, the distance is 18.4 meters and the speed is 44.4448 m/s.\[ \text{Time} = \frac{18.4}{44.4448} = 0.414 \text{ seconds} \]

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Key Concepts

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

Speed Conversion
When tackling problems involving different units, speed conversion is crucial. Imagine being in a situation where you're given speed in kilometers per hour, but you need to work in meters per second. To make this conversion, you use a simple unit rate: 1 km/h is equivalent to approximately 0.27778 meters per second. So, what does this mean practically?
For instance, if you're measuring the speed of a baseball at 160 km/h, you multiply by 0.27778 to convert it into meters per second. This gives you:
\[ 160 \times 0.27778 = 44.4448 \text{ m/s} \] This conversion is vital because it aligns the units with the distance, simplifying further calculations. It ensures that every value you're working with is harmonious and makes calculating time or determining any other outcomes straightforward.
Distance Calculation
Although the original exercise gives the distance directly, understanding distance calculation is vital for mastering projectile motion problems. Distance, in most basic terms, can be thought of as the total stretch or expanse an object covers during its motion. The given distance, like the 18.4 meters in our exercise, is how far the baseball travels. In scenarios where this is not provided, you typically calculate it using other known values and applying appropriate kinematic formulas.
Distance calculations might be used to determine how different variables such as velocity or time affect how far an object travels. In multi-part projects or real-life scenarios, these calculations help predict outcomes and plan more effectively.
Time Calculation
Calculating time is a critical component in many physics problems, especially in projectile motion. Time tells us how long an object, like a baseball, will take to reach a specific point. The key formula here is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] Divide the distance by the converted speed to get the time in seconds. From our exercise, with a distance of 18.4 meters and speed of 44.4448 meters per second, this comes to:
\[ \text{Time} = \frac{18.4}{44.4448} = 0.414 \text{ seconds} \] Use this calculation method in various scenarios from sports to transport dynamics, enabling accurate prediction of how different elements interact in motion over time.

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Most popular questions from this chapter

A motorcycle is moving at \(30 \mathrm{~m} / \mathrm{s}\) when the rider applies the brakes, giving the motorcycle a constant deceleration. During the \(3.0 \mathrm{~s}\) interval immediately after braking begins, the speed decreases to \(15 \mathrm{~m} / \mathrm{s} .\) What distance does the motorcycle travel from the instant braking begins until the motorcycle stops?

Compute your averagc velocity in the following two cascs: (a) You walk \(73.2 \mathrm{~m}\) at a speed of \(1.22 \mathrm{~m} / \mathrm{s}\) and then run \(73.2 \mathrm{~m}\) at a speed of \(3.05 \mathrm{~m} / \mathrm{s}\) along a straight track. (b) You walk for \(1.00 \mathrm{~min}\) at a speed of \(1.22 \mathrm{~m} / \mathrm{s}\) and then run for \(1.00 \mathrm{~min}\) at \(3.05 \mathrm{~m} / \mathrm{s}\) along a straight track. (c) Graph \(x\) versus \(t\) for both cases and indicate how the average velocity is found on the graph.

The sport with the fastest moving ball is jai alai, where measured speeds have reached \(303 \mathrm{~km} / \mathrm{h}\). If a professional jai alai player faces a ball at that speed and involuntarily blinks, he blacks out the scene for \(100 \mathrm{~ms}\). How far does the ball move during the blackout?

A rock is dropped (from rest) from the top of a \(60-\mathrm{m}\) -tall building. How far above the ground is the rock \(1.2 \mathrm{~s}\) before it reaches the ground?

Two particles move along an \(x\) axis. The position of particle 1 is given by \(x=6.00 t^{2}+3.00 t+2.00\) (in meters and seconds); the acceleration of particle 2 is given by \(a=-8.00 t\) (in meters per second squared and seconds) and, at \(t=0,\) its velocity is \(20 \mathrm{~m} / \mathrm{s}\). When the velocities of the particles match, what is their velocity?

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