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?

Short answer

Minimum return rate is 20.0 cm/s.

Step-by-step solution

Calculate time to reach full scale

First, determine how long it takes for the pointer to traverse the full 12.0 cm scale at a rate of 10.0 cm/s. This is done by dividing the distance to be covered by the rate of travel. Time to reach full scale, t1 = Distance / Rate = 12.0 cm / 10.0 cm/s.

Calculate remaining time for return trip

Since the total time for the round trip is 2.00 seconds, subtract the time to reach the full scale (t1) from the total time to find the remaining time for the pointer to return to zero. Remaining time, t2 = Total time - Time to reach full scale.

Calculate minimum return rate

To calculate the minimum return rate necessary for the pointer to make it back to the start in the remaining time (t2), divide the full scale distance by the remaining time. Minimum return rate = Full scale distance / Remaining time.

Key concepts

Relative Speed

Relative speed is a concept used to describe the speed of an object in relation to another moving object. It is particularly useful in scenarios where two objects are moving toward each other or in the same direction. In the context of our exercise, if we had another pointer on the meter traveling in the opposite direction to our original pointer, the relative speed would be the sum of their individual speeds when calculating how quickly they meet.

However, if they were moving in the same direction, we would instead subtract the speed of the slower pointer from the speed of the faster one to get the relative speed. Understanding this is crucial when considering moving objects and plays a significant role in problems involving races, overtaking, and even physics problems involving relative motion.

Distance-Time Relationship

The distance-time relationship is a fundamental concept in understanding motion. It refers to the direct proportionality between the distance traveled and the time taken when speed is constant. Essentially, distance equals speed multiplied by time \(d = st\), and can be visualized with the formula \(t = d/s\) when solving for time, as seen in the step by step solution provided for the exercise.

This relationship helps us determine how long it takes an object to cover a certain distance at a steady speed, or conversely, how far an object can travel in a given time frame. Grasping this concept is vital not only for solving basic motion problems but also for understanding more complex dynamics in everyday scenarios, such as travel planning or athletic training.

Round Trip Calculation

A round trip calculation involves finding the total time or speed for a journey to a certain point and back to the starting point. In the solved exercise, the pointer needs to cover the distance twice, once forward and once in return, within a specific total time frame.

To determine how long it will take or how fast the object must move, we separate the round trip into two single trips: the 'outward' leg and the 'return' leg. Just as we calculated for the pointer, we subtract the time taken for the outward leg from the total time to find the time remaining for the return leg. Understanding the principle of round trip calculation is essential in many real-life applications ranging from transportation logistics to athletic event planning, where the return time or speed may differ from the outward ones.

Average Speed Calculation

Average speed calculation is the process of determining the overall speed of an object over a distance where it may have traveled at varying speeds. It's different from the instantaneous speed, which is the speed at any given moment of the trip. The average speed is calculated by dividing the total distance traveled by the total time taken \(v_{avg} = d_{total}/t_{total}\).

For our exercise, if the pointer moved at different speeds during its trip, the average speed would not be the simple mean of the two speeds. Instead, it would be based on the total distance (in this case, double the scale length since it's a round trip) divided by the total time for the journey. This concept is widely applied in everyday life, such as calculating the average speed of a car trip that involves varied speed limits.