Problem 109 - Fundamentals of Physics

Question

An automobile driver increases the speed at a constant rate from \(25 \mathrm{~km}\) 'h to \(55 \mathrm{~km} / \mathrm{h}\) in \(0.50 \mathrm{~min}\). A bicycle rider speeds up at a constant rate from rest to \(30 \mathrm{~km} / \mathrm{h}\) in \(0.50 \mathrm{~min}\). What are the magnitudcs of (a) the driver's acceleration and (b) the rider's acceleration?

Step-by-Step Solution

Answer

Both have an acceleration of approximately 0.28 m/s².

Step 1: Convert Time to Seconds

First, convert the time from minutes to seconds because the standard unit for acceleration is meters per second squared (m/s²). Since there are 60 seconds in a minute, \(0.50\) minutes is equivalent to \(0.50 \times 60 = 30\) seconds.

Step 2: Convert Speeds to Meters per Second

Convert speeds from kilometers per hour to meters per second using the conversion factor \(1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s}.\)- For the automobile driver: \[ 25 \, \text{km/h} = 25 \times \frac{1000}{3600} \approx 6.94 \, \text{m/s}\] \[ 55 \, \text{km/h} = 55 \times \frac{1000}{3600} \approx 15.28 \, \text{m/s}\]- For the bicycle rider: \[ 0 \, \text{km/h} = 0 \times \frac{1000}{3600} = 0 \, \text{m/s}\] \[ 30 \, \text{km/h} = 30 \times \frac{1000}{3600} \approx 8.33 \, \text{m/s}\]

Step 3: Calculate Driver's Acceleration

Acceleration is the change in velocity divided by the time taken for the change. Use the formula:\[ a = \frac{v_f - v_i}{t} \]For the automobile driver:\[ v_f = 15.28 \, \text{m/s}, \quad v_i = 6.94 \, \text{m/s}, \quad t = 30 \, \text{s} \]\[ a = \frac{15.28 - 6.94}{30} = \frac{8.34}{30} \approx 0.28 \, \text{m/s}^2 \]

Step 4: Calculate Rider's Acceleration

Again, use the acceleration formula:\[ a = \frac{v_f - v_i}{t} \]For the bicycle rider:\[ v_f = 8.33 \, \text{m/s}, \quad v_i = 0 \, \text{m/s}, \quad t = 30 \, \text{s} \]\[ a = \frac{8.33 - 0}{30} = \frac{8.33}{30} \approx 0.28 \, \text{m/s}^2 \]

Key Concepts

Understanding Acceleration

Acceleration is a fundamental concept in kinematics. It describes how quickly the velocity of an object increases or decreases. In simpler terms, it's the rate at which you speed up or slow down. Acceleration is calculated using the formula:

• \( a = \frac{v_f - v_i}{t} \)

Here,

• \( a \) represents acceleration,
• \( v_f \) is the final velocity,
• \( v_i \) is the initial velocity,
• \( t \) is the time over which the change occurs.

This formula tells us that acceleration occurs when there's a change in velocity over a specific period. In the given problem, both the car and the bicycle accelerate smoothly, meaning their velocities change at a constant rate. Thus, both drivers experience a constant acceleration of approximately \( 0.28 \, \text{m/s}^2 \).Remember, acceleration is measured in meters per second squared (m/s²), which indicates how much the speed changes every second.

The Role of Velocity in Motion

Velocity plays a crucial role when understanding motion. Unlike acceleration, which refers to the rate of change of velocity, velocity itself describes the speed in a particular direction. It gives us two important pieces of information: how fast an object is moving and in what direction.In the context of this problem, the car starts at an initial velocity of \( 25 \, \text{km/h} \) and speeds up to \( 55 \, \text{km/h} \), whereas the bicycle accelerates from a stop (\( 0 \, \text{km/h} \)) up to \( 30 \, \text{km/h} \). Thanks to their velocities, we can better understand their change in speed over time, which ultimately leads us to find their acceleration. Ensuring that velocity is correctly converted to meters per second (from kilometers per hour) aligns with the standard calculation metrics.Remember, without a change in velocity, there can be no acceleration.

Importance of Unit Conversion in Physics

Unit conversion is an essential skill in physics to ensure that calculations are accurate and meaningful. Changes in measurement units can offer convenience in comprehending quantities in a certain context. When considering kinematics, it's critical to convert all measurements to compatible units to apply physical formulas correctly.In this exercise, converting kilometers per hour (\(\text{km/h}\)) to meters per second (\(\text{m/s}\)) and minutes to seconds helps in applying the kinematics formula effectively.

• The conversion factor for speed: \( 1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s} \).
• Time conversion: 1 minute equals 60 seconds.

These conversions allow for precise computation and ensure that the conceptual understanding translates into accurate quantitative analysis. Without this diligent conversion step, calculations could yield incorrect results due to incompatible units.