Problem 37 - Algebra 2

Question

Physics The acceleration of an object is a measure of how much its velocity changes in a given period of time. acceleration \(=\frac{\text { final velocity }-\text { initial velocity }}{\text { time }}\) Suppose you are riding a bicycle at 6 \(\mathrm{m} / \mathrm{s}\) . You step hard on the pedals and increase your speed to 12 \(\mathrm{m} / \mathrm{s}\) in about 5 \(\mathrm{s}\) . a. Find your acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) . b. A sedan can go from 0 to 60 \(\mathrm{mi} / \mathrm{h}\) in about 10 s. What is the acceleration in \(\mathrm{m} / \mathrm{s}^{2} ?(\text { Hints: } 1 \mathrm{mi} \approx 1609 \mathrm{m} ; 1 \mathrm{h}=3600 \mathrm{s} .)\)

Step-by-Step Solution

Answer

The acceleration of the bicycle is 1.2 m/s², while the acceleration of the sedan is 2.68 m/s².

Step 1: Calculate the bicycle acceleration

From the problem, we are told the initial velocity (\(u\)) is 6 m/s, the final velocity (\(v\)) is 12 m/s and the time taken (\(t\)) is 5s. Substitute these values into the formula for acceleration (\(a\)): \(a = \frac{v - u}{t}\). Therefore, the acceleration of the bicycle equals: \(a_{bicycle} = \frac{12 m/s - 6 m/s}{5 s} = 1.2 m/s².\)

Step 2: Convert units for sedan

For the sedan, we need to convert the speed from miles per hour to meters per second, and the time from hours to seconds. Use the given conversion factors: \(1 mi \approx 1609 m\) and \(1 h = 3600 s\). Converting gives us a final speed (\(v_{sedan}\)) of \(60 mi/h * \frac{1609 m}{1 mi} * \frac{1 h}{3600 s} = 26.82 m/s\) and a time (\(t_{sedan}\)) of 10 s.

Step 3: Calculate the sedan acceleration

Now, apply the formula for acceleration, using the initial velocity (\(u_{sedan}\)) of 0 m/s (since the car starts from rest), the final velocity (\(v_{sedan}\)) of 26.82 m/s and the time (\(t_{sedan}\) of 10s. Calculate using \(a_{sedan} = \frac{v_{sedan} - u_{sedan}}{t_{sedan}} \) to find: \(a_{sedan} = \frac{26.82 m/s - 0 m/s}{10 s} = 2.68 m/s²\).

Key Concepts

Velocity

Velocity is a fundamental concept in physics. It describes how fast something is moving in a specific direction. Unlike speed, which tells us how fast an object is going, velocity includes direction, making it a vector quantity. In our exercise, we encountered two states of velocity: initial and final. The initial velocity is where the object starts, and the final velocity is where it ends after acceleration.
To determine how velocity changes, we measure the rate of this change over time, which is termed as acceleration. The velocity component in this problem is measured in meters per second (m/s), allowing us to easily calculate acceleration. To find acceleration, we need both the initial and final velocities along with the time taken for the change.

Unit Conversion

Unit conversion is a crucial step in solving physics problems because it ensures we work with compatible units. Often, we're faced with different measurement systems and need to bring them to a common unit.
In physics, particularly when working with velocity and acceleration, we often convert units from miles per hour (mi/h) to meters per second (m/s). In our exercise, the sedan's speed was originally given in mi/h. The key steps in unit conversion include:

• Convert miles to meters using the conversion factor ( 1 mi = 1609 m).
• Convert hours to seconds using the factor ( 1 h = 3600 s).

This method ensures that all calculations are performed in the International System of Units (SI), which is preferred in scientific contexts.

Physics Problem Solving

Successfully solving physics problems involves a structured approach. Begin by clearly understanding the problem statement: know what information is given and what needs to be found. Let's outline our approach with the exercise:

• Identify given values and required units.
• Execute necessary unit conversions.
• Utilize relevant formulas to find unknowns.

In our acceleration problem, this involves substituting known values into the formula for acceleration: \(a = \frac{v - u}{t}\) where \(v\) is final velocity, \(u\) is initial velocity, and \(t\) is time.
Breaking the problem into parts and proceeding step by step helps simplify complex problems, ensuring all aspects are considered.

Rate of Change

The rate of change is vital in understanding how quickly something changes over a given period. In physics, when we talk about rate of change concerning motion, we're generally talking about acceleration. Acceleration tells us how quickly velocity changes over time.
The concept is straightforward: more rapid changes in velocity mean higher acceleration. This can be witnessed if you compare a bicycle speeding up with a car accelerating. The car's larger change in velocity over a shorter time results in greater acceleration.
To calculate this, use the formula: \(a = \frac{v - u}{t}\). You subtract the initial velocity from the final velocity to find how much the velocity has increased. Divide by the time it takes for this change to determine the acceleration rate, expressed in meters per second squared \(m/s^2\). This measure provides insight into the dynamics of any motion.