Problem 2 - Technical Mathematics with Calculus

Question

The pointer of a certain meter can travel to the right at the rate of \(10.0 \mathrm{cm} / \mathrm{s}\) What must be the minimum return rate if the total time for the pointer to traverse the full 12.0 -cm scale and return to zero must not exceed 2.00 seconds?

Step-by-Step Solution

Answer

The minimum return rate is 20.0 cm/s.

Step 1: Calculating Time to Traverse Scale

Determine the time the pointer takes to traverse the full scale to the right. Since the scale is 12.0 cm, and the rate of travel is 10.0 cm/s, the time taken to travel to the right is found by dividing the distance by the speed: Time = Distance / Speed = 12.0 cm / 10.0 cm/s.

Step 2: Finding Time Remaining for Return

Subtract the time taken to travel to the right from the total time allowed for the round trip. The pointer needs to complete the round trip in 2.00 seconds, so the remaining time is Total Time - Time to Traverse = 2.00 s - (12.0 cm / 10.0 cm/s).

Step 3: Calculating Minimum Return Rate

With the remaining time calculated, find the minimum return rate by dividing the scale length (12.0 cm) by the remaining time. This will give the minimum speed required to return to zero in the time allowed.

Key Concepts

Uniform Motion

Understanding uniform motion is fundamental in solving kinematics problems. Uniform motion refers to the movement of an object at a constant speed in a straight line. In this context, uniform means 'unchanging' or 'steady', meaning the rate of travel does not vary with time.

For an object in uniform motion, such as the pointer of the meter in the exercise, it means that its speed stays at 10.0 cm/s as it travels to the right. There are no accelerations or decelerations during this phase of its journey. It allows for the use of a simple formula to calculate distance, speed (rate), or time: Distance = Speed x Time. If any two variables are known, this equation can quickly give us the third, which is essential for solving problems involving uniform motion.

Rate of Travel

The rate of travel is simply the speed at which an object is moving, usually expressed in terms such as centimeters per second (cm/s), kilometers per hour (km/h), or miles per hour (mph). It tells us how much distance the object covers in a given amount of time.

To calculate the speed of the meter's pointer, we would use the formula Speed = Distance / Time. In the step-by-step solution, the pointer’s rate of travel to the right is given as 10.0 cm/s. To find the minimum return rate, we also use the rate of travel concept, but in reverse: by calculating the necessary speed based on the allowed time and the distance that needs to be covered.

Time Calculation

Calculating time correctly is crucial in kinematics and physics mathematics. The formula Time = Distance / Speed is derived from the fundamental definition of speed and is used when the speed and distance are known, but the time needs to be determined.

In our example, this formula helps us calculate the time it takes for the pointer of the meter to travel to the right. The solution reinforces the concept by subtracting the travel time from the total allowed time, thereby determining how much time is remaining for the pointer to return. This calculation is essential in ensuring the complete round trip does not surpass the 2-second requirement.

Physics Mathematics

Physics mathematics involves applying mathematical methods to solve physics problems. It incorporates formulas, algebra, calculus, and other mathematical concepts to analyze physical scenarios. This problem showcases the application of algebra, where the known and given values are plugged into applicable formulas to solve for unknown variables.

In the exercise, algebra is used to rearrange the formula and calculate the required return rate of the pointer. We first calculate the time taken for one-way motion, then determine the remaining time for the return trip, and finally use this information to work out the minimum return rate. Not only does this strengthen the understanding of algebraic manipulation, but it also provides a practical application of basic physical concepts in a real-world context.