Problem 88 - Calculus and its applications

Question

A motorcycle accelerates at a constant rate from 0 to \(50 \mathrm{mph}\) in \(15 \mathrm{sec} .\) How far does it travel during that time?

Step-by-Step Solution

Answer

The motorcycle travels 550.125 feet during the 15 seconds.

Step 1: Convert Speeds to Consistent Units

First, convert the speed from miles per hour (mph) to feet per second (ft/s) since time is given in seconds. The conversion factor is 1 mph = 1.467 ft/s. So, 50 mph is equivalent to \(50 \times 1.467 = 73.35 \text{ ft/s}\).

Step 2: Calculate Average Speed

The initial speed is 0, and the final speed is 73.35 ft/s. The average speed during constant acceleration is given by the average of the initial and final speeds: \(\text{Average Speed} = \frac{0 + 73.35}{2} = 36.675 \text{ ft/s}\).

Step 3: Calculate Distance Traveled

Using the formula for distance \(\text{Distance} = \text{Average Speed} \times \text{Time}\), and substituting the known values: \(\text{Distance} = 36.675 \text{ ft/s} \times 15 \text{ s} = 550.125 \text{ ft}\). Thus, the motorcycle travels 550.125 feet.

Key Concepts

Unit Conversion

Understanding unit conversion is vital in physics as it allows us to use consistent units across calculations. Imagine speaking a foreign language; unit conversion is like translating words into a language you understand. In the given exercise, speed was originally presented in miles per hour (mph), but the calculations required feet per second (ft/s). Thus, we needed to convert mph into ft/s.
This is done using a known conversion factor: 1 mph equals 1.467 ft/s. Knowing this, you can convert the final speed of the motorcycle by multiplying:

• Convert 50 mph to ft/s by calculating: \[50 \times 1.467 = 73.35 \text{ ft/s}\]

This conversion is essential for consistent and accurate calculations when blending different units of measurement. Getting comfortable with unit conversions makes tackling physics problems much easier.

Constant Acceleration

Constant acceleration is when an object's velocity changes at a steady rate over time. Think of it like a car steadily pressing the gas pedal, causing its speed to increase evenly rather than suddenly. In our motorcycle example, this means increasing speed from 0 mph to 50 mph at a consistent rate.
Because the acceleration is constant, it allows us to compute average speed in a straightforward way. The average speed is calculated by taking the mean of the initial speed and the final speed:

• Initial speed = 0 ft/s
• Final speed = 73.35 ft/s
• Average Speed = \(\frac{0 + 73.35}{2} = 36.675 \text{ ft/s}\)

This average speed represents the motorcycle's overall speed throughout its acceleration period, which simplifies the calculation of distance traveled.

Distance Formula

The distance formula is like a map guiding us to where an object goes globally. With the average speed known, we can now calculate the total distance the motorcycle travels during acceleration. The key formula here is:

• Distance = Average Speed \(\times\) Time

Using the values from the problem and what we've derived:

• Average Speed = 36.675 ft/s
• Time = 15 s
• Distance traveled = \(36.675 \text{ ft/s} \times 15 \text{ s} = 550.125 \text{ ft}\)

The answer tells us the motorcycle covers 550.125 feet in 15 seconds, all thanks to the steady acceleration over that time. It illustrates how easy it is to map an object's path when its speed changes at a constant rate.