Two runners travel along the same straight path. They start at the same time, and they end at the same time, but at the halfway mark, they have different instantaneous velocities. Is it possible for them to have the same average velocity for the trip?
They wouldn't, right? Because instantaneous velocity is just the midpoint of the average velocity and if they start and end at the same time, then shouldn't their instantenous velocity be at the same time? Unless one sped up and then slowed down while another slowed down and sped up? But shouldn't their velocities remain constant?
of course their average velocities are the same
avg velocity = totaldistance / total time
Thanks. I guess I was just overthinking it.
It is indeed possible for the two runners to have the same average velocity for the trip, even if they have different instantaneous velocities at the halfway mark. The key point to understand is that instantaneous velocity measures the speed of an object at a particular moment in time, while average velocity considers the total displacement and total time taken for the entire trip.
Let's consider an example to illustrate this concept. Suppose Runner A starts at point A and runs at a constant speed until reaching point B, where the halfway mark is located. Runner B, on the other hand, starts at point B and runs at a different speed until reaching point A. At the halfway mark, their instantaneous velocities will be different because they are at different points on the path at that moment.
However, what matters for average velocity is the total displacement (change in position) and the total time taken for the trip. Since both runners start and end at the same time, their total time will be the same. If they cover equal displacements but in opposite directions (which is the case in our example), then their average velocities will be the same, regardless of the variations in their instantaneous velocities.
So, in summary, it is possible for the runners to have the same average velocity for the trip, despite having different instantaneous velocities at the halfway mark.
To determine if it's possible for the two runners to have the same average velocity for the trip, let's break down the concepts involved.
Average velocity is calculated by dividing the total displacement of an object by the total time it takes to cover that displacement. It is the overall change in position divided by the total time.
Instantaneous velocity, on the other hand, refers to the velocity of an object at a specific point in time. It is the limit of the average velocity as the time interval approaches zero.
Now, let's consider the scenario described. The two runners start and end at the same time but have different instantaneous velocities at the halfway mark. This means that their velocities are not constant throughout the entire trip.
To determine if they can have the same average velocity, we need to consider the displacement and time for each runner.
If the runners cover the same total displacement during the trip and take the same amount of time, then they will have the same average velocity. However, for them to have different instantaneous velocities at the halfway mark, their velocities must change during the trip.
Therefore, it's possible for the two runners to have the same average velocity if they cover equal distances but experience changes in velocity, such as one speeding up and then slowing down while the other slows down and then speeds up.
In summary, as long as the runners cover the same distance and take the same amount of time, they can have the same average velocity, even if their instantaneous velocities are different at specific points along the way.