Chapter 2: Problem 20
An automobile driver on a straight road increases the speed at a constant ratefrom \(25 \mathrm{~km} / \mathrm{h}\) to \(55 \mathrm{~km} / \mathrm{h}\) in \(0.50\mathrm{~min}\). A bicycle rider on a straight road speeds up at a constantrate from rest to \(30 \mathrm{~km} / \mathrm{h}\) in \(0.50 \mathrm{~min} .\)Calculate their accelerations.
Short Answer
Both the automobile and the bicycle have the same acceleration: 0.278 m/s².
Step by step solution
01
- Convert time to seconds
First, convert the time from minutes to seconds because the standard unit for calculating acceleration is meters per second squared (\text{m/s}^2). Given time: 0.50 minutes. We perform the conversion as follows: \[ 0.50 \text{ minutes} \times 60 \text{ seconds per minute} = 30 \text{ seconds} \]
02
- Convert velocities to meters per second
Convert the velocities from \text{km/h} to \text{m/s} using the conversion factor, where 1 \text{ km/h} = \(\frac{1}{3.6}\) \text{ m/s}. For the automobile: Initial speed: \[ 25 \text{ km/h} \times \frac{1}{3.6} = 6.94 \text{ m/s} \] Final speed: \[ 55 \text{ km/h} \times \frac{1}{3.6} = 15.28 \text{ m/s} \] For the bicycle: Initial speed: \[ 0 \text{ km/h} = 0 \text{ m/s} \] Final speed: \[ 30 \text{ km/h} \times \frac{1}{3.6} = 8.33 \text{ m/s} \]
03
- Calculate acceleration for the automobile
Use the formula for constant acceleration: \( a = \frac{v_f - v_i}{t} \) For the automobile: Initial speed: 6.94 \text{ m/s} Final speed: 15.28 \text{ m/s} Time: 30 \text{ seconds} \[ a = \frac{15.28 \text{ m/s} - 6.94 \text{ m/s}}{30 \text{ s}} = \frac{8.34 \text{ m/s}}{30 \text{ s}} = 0.278 \text{ m/s}^2 \]
04
- Calculate acceleration for the bicycle
Use the formula for constant acceleration: \( a = \frac{v_f - v_i}{t} \) For the bicycle: Initial speed: 0 \text{ m/s} Final speed: 8.33 \text{ m/s} Time: 30 \text{ seconds} \[ a = \frac{8.33 \text{ m/s}}{30 \text{ s}} = 0.278 \text{ m/s}^2 \]
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
constant acceleration
When we talk about constant acceleration, we mean that the rate of change of velocity remains the same over time. Imagine a car that speeds up from 25 km/h to 55 km/h in a smooth, even manner.
This means it doesn't suddenly jump in speed or slow down; it accelerates at a steady pace.
To find acceleration, we use the formula: \( a = \frac{v_f - v_i}{t} \). Here, \( a \) is acceleration, \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( t \) is time.
In our example, both the car and the bike accelerate consistently, which makes the calculations straightforward.
Understanding this concept is crucial as it applies not only to homework problems but also to real-life driving and biking scenarios.
velocity conversion
We often use different units to measure speed, like kilometers per hour (km/h) or meters per second (m/s).
To solve physics problems, converting these units accurately is necessary. Usually, we need speed in meters per second for calculations.
The rule to remember is that \( 1 \) km/h equals \( \frac{1}{3.6} \) m/s. In the exercise, we converted 25 km/h to 6.94 m/s and 55 km/h to 15.28 m/s for the car and 30 km/h to 8.33 m/s for the bicycle.
Always do this step carefully to avoid mistakes in further calculations.
A quick way to remember is to divide speeds in km/h by 3.6, and you'll have them in m/s!
unit conversion
Unit conversion is a vital skill in physics to ensure that all measurements are in the correct units for formulas.
In our example, we converted time from minutes to seconds because the standard unit for time in physics is seconds when calculating acceleration.
The conversion is simple: \( \text{minutes} \times 60 = \text{seconds} \). Thus, 0.50 minutes becomes 30 seconds.
For velocity, converting from kilometers per hour to meters per second involves multiplying by \( \frac{1}{3.6} \).
Master these conversions to simplify and solve physics problems accurately.
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