Solving a Displacement-Time Graph with Sharp Turnings

[Listin]

I got a problem described by a displacement time graph. It has sharp turnings at 2 points ( and sucessives) and the question is to convert the graph to velocity time graph. Since sharp pionts are not differentiable how it can be drawn ?

 

[Fewmet]

I got a problem described by a displacement time graph. It has sharp turnings at 2 points ( and sucessives) and the question is to convert the graph to velocity time graph. Since sharp pionts are not differentiable how it can be drawn ?

This is a little tricky to answer because in a class room setting it depends on your professor's perspective.

In the real world there cannot be perfectly sharp points as that implies an infinite rate of change of velocity and therefor infinite force being applied.

Is the graph produced from real data? If so, the sharp point is either a matter of infrequent sampling missing the points that would clearly define the change of velocity there, or it could be that the scale of the graph is not sufficiently fine to show the rapid change.

Could the professor be trying trying teach you this by giving you an apparently impossible graph?

If you want to test the graph as not necessarily bound by natural law (i.e., as a mathematical ideal), could use the math notation of inserting a circle in the velocity time graph to in indicate a discontinuity.

 

[HallsofIvy]

If there are sharp corners in the displacement graph, the velocity graph will not be continuous there. For example, if the displacement graph is straight line from (0, 0) to (5, 10) and then changes to the straight line from (5, 10) to (10, 10). The velocity will be the constant (10- 0)/(5- 0)= 2 from 0 to 5, then the constant (10-10)/(10-5)= 0 from 5 to 10. The velocity graph will be the horizontal line from (0, 2) to (5, 2), then the horizontal line from (5, 0) to (10, 0).

 

[nasu]

It is quite common in high school physics to show graphs of the motion (position-time and velocity-time) made from straight line segments connected at various angles.
Usually the questions related to these graphs regard the values at various points on the segments or average values for a segment, in which case there is no problem.
If they ask for instantaneous values at these connection points, then there is a problem.

If the question is just to convert to v-t graph then they may expect you calculate the average velocity on each segment and build another graph made from straight lines.

 

[PJC01]

Thank you for bringing this interesting problem to my attention. As a scientist, my approach would be to first analyze the displacement-time graph to understand the underlying motion. Sharp turnings in a displacement-time graph indicate a change in the direction of motion, which could be due to a change in velocity or acceleration.

To convert the graph to a velocity-time graph, we need to determine the velocity at each point of the sharp turnings. Since these points are not differentiable, we cannot use the traditional method of finding the slope of the tangent line. Instead, we can use the concept of average velocity over a small time interval.

We can divide the time interval between two successive sharp turnings into smaller intervals and calculate the average velocity for each interval. This will give us a series of data points that can be plotted on a velocity-time graph. The smaller the time interval, the more accurate the results will be.

It is important to note that the velocity-time graph obtained using this method will not be a smooth curve, but rather a series of data points connected by straight lines. This is because the sharp turnings in the displacement-time graph indicate abrupt changes in velocity, which will be reflected in the velocity-time graph.

In conclusion, while sharp turnings in a displacement-time graph do present a challenge in converting it to a velocity-time graph, it is not impossible. By using the concept of average velocity and dividing the time interval into smaller intervals, we can accurately plot the velocity-time graph. It would also be helpful to discuss this problem with other scientists and collaborate on potential solutions.