When would instantaneous speed be the same as average speed?
would it be
when the speed is constant?
when the position changes linearly with time
or would it be in the limit when the time interval goes to zero?
I thought it would be when the speed is constant, but i'm not sure...
The correct answer is in the limit when the time interval goes to zero. Let me explain why.
Average speed is defined as the total distance traveled divided by the total time taken. Mathematically, it can be represented as:
Average speed = (Total distance traveled) / (Total time taken)
Instantaneous speed, on the other hand, refers to the speed of an object at a specific moment in time. It can be thought of as the speed of an object at a particular instant.
To understand when instantaneous speed is the same as average speed, let's consider different scenarios:
1. When the speed is constant: In this case, the object is moving at a constant speed throughout its journey. As a result, the average speed and instantaneous speed will be the same at any given time. So, yes, the statement holds true for a constant speed.
2. When the position changes linearly with time: If the object's position changes linearly with time, it means its speed remains constant over a given period. In this situation, as we discussed earlier, the instantaneous speed at any instant will be the same as the average speed.
3. In the limit when the time interval goes to zero: This is the most general condition when instantaneous speed is always equal to average speed. When we calculate average speed, we divide the total distance traveled by the total time taken. As the time interval becomes infinitesimally small (going to zero), the average speed calculated over that interval approaches the instantaneous speed at that specific moment. This concept is based on the derivative in calculus, where the average speed can be seen as the derivative of distance with respect to time.
Therefore, the correct answer is that instantaneous speed is the same as average speed in the limit when the time interval goes to zero.