Problem 81 - Calculus and its applications

Question

A motorcycle accelerates at a constant rate from \(0 \mathrm{mph}(v(0)=0)\) to \(60 \mathrm{mph}\) in \(15 \mathrm{sec}\). How far has it traveled after 15 sec? (Hint: Convert seconds to hours.)

Step-by-Step Solution

Answer

The motorcycle traveled \(\frac{1}{8}\) miles in 15 seconds.

Step 1: Identify Known Values

We know the motorcycle starts at rest, so the initial velocity is \(v(0) = 0 \text{ mph}\). The final velocity is \(60 \text{ mph}\), and the time duration is \(15 \text{ seconds}\).

Step 2: Convert Time to Hours

We convert the time from seconds to hours since our velocities are in miles per hour. There are 3600 seconds in an hour, so \(15 \text{ sec} = \frac{15}{3600} \text{ hours} = \frac{1}{240} \text{ hours}\).

Step 3: Calculate Average Velocity

The motorcycle undergoes constant acceleration, so the average velocity is the mean of initial and final velocities: \(\text{Average Velocity} = \frac{v(0) + v(t)}{2} = \frac{0 + 60}{2} = 30 \text{ mph}\).

Step 4: Calculate Distance Traveled

Using the formula \(\text{Distance} = \text{Average Velocity} \times \text{Time}\), we substitute the known values: \(\text{Distance} = 30 \text{ mph} \times \frac{1}{240} \text{ hours} = \frac{30}{240} \text{ miles} = \frac{1}{8} \text{ miles}\).

Step 5: Finalize the Solution

The result from our calculation shows us that the motorcycle traveled \(\frac{1}{8} \text{ miles}\).

Key Concepts

Average Velocity

To understand average velocity in the context of constant acceleration, think of it as the middle point between the starting and ending velocities. When a vehicle accelerates at a constant rate, the velocities change evenly over time. This allows us to calculate the average simply as the arithmetic mean of the initial and final velocities. For example, in the given motorcycle problem:

• The initial velocity is 0 mph (because it starts from rest).
• The final velocity is 60 mph (as it reaches that speed).

The average velocity, therefore, is calculated as:\[\text{Average Velocity} = \frac{v(0) + v(t)}{2} = \frac{0 + 60}{2} = 30 \text{ mph}\]This concept ensures that even though the speed varies, we can determine an overall speed for calculations.

Distance Calculation

Calculating the distance traveled under constant acceleration involves combining average velocity with the time over which the motion occurs. Once we have the average velocity, distance is straightforward to find using the formula:\[ \text{Distance} = \text{Average Velocity} \times \text{Time} \]In our example:

• The average velocity is found to be 30 mph.
• The time duration of the motorcycle's motion, after conversion to hours, is \( \frac{1}{240} \) hours.

Thus, the distance calculation becomes:\[\text{Distance} = 30 \text{ mph} \times \frac{1}{240} \text{ hours} = \frac{30}{240} \text{ miles} = \frac{1}{8} \text{ miles}\]This formula and method give us the total distance traveled by accounting for varying speed over the course of travel.

Unit Conversion

Unit conversion plays a crucial role in physics problems, especially when there is a mismatch in units, like time in seconds and velocity in mph. Accurate conversion ensures calculations are consistent and meaningful. Here are simple steps to convert time from seconds to hours, as needed:

• Recognize there are 3600 seconds in one hour.
• Convert the time by dividing the number of seconds by 3600.

For instance, in the motorcycle example:\[ 15 \text{ seconds} = \frac{15}{3600} \text{ hours} = \frac{1}{240} \text{ hours} \]This conversion allows the calculation of distance using miles per hour (mph) and ensures that velocity and time dimensions align correctly for the formula. Proper unit conversion is a vital step in solving problems accurately and effectively.