In my physics class we are learning about velocity vs. time graphs and position vs. time graphs for an object. We have learned that if the object is moving with constant velocity (this would be a diagonal line on the position vs. time graph) that the velocity vs. time graph would be a horizontal line. We have also learned that if the object is accelerating and is not moving with constant velocity (a curved position vs. time graph) that the velocity vs. time graph would be a diagonal line. However, what would the graph of the Position vs. Time graph look like if the Velocity vs. Time graph was a curve?
[Physics] What would the position vs. time graph look like if the velocity vs. time graph was a curve
accelerationkinematicsspeedtimevelocity
Related Solutions
As you said, the next derivative of the velocity with respect to time is the acceleration. And the acceleration could in principle have a step somewhere due to a force starting to act on the object.
Since only data you have is this table, you don't need to connect the points and speculate on if its graph is linear or not.You can't really know its properties with this much information. Each interval can either be linear or nonlinear on its own. Therefore, you can just leave it like this:
If you really want to sketch the velocity-time graph(again you don't have to with just this much information), you can find the average velocity for each interval and sketch it accordingly. However, take these points into consideration while sketching it:
- What you essentially need to accomplish is to make the area under the graph at each interval to be equal to $\Delta x$ of each interval.
- The average velocity for each interval would be $2.2$ $m/s$, $-1.4$ $m/s$, $-3.8$ $m/s$, $-3.7$ $m/s$, $-1.6$ $m/s$ respectively. However, you can't really set these values in the graph since you can't know at what point these velocities are reached. You need to know instantaneous velocity of the object at each time to accurately sketch the graph.(which will require $x(t)$ function)
- The graph will just be an approximation and probably not even a close one.
Best Answer
Let’s say the speed was described by a curve that is itself described by some function of time $v = f(t)$. If you know what the initial position $x_0$ is, then the curve of position vs. time would be described by
$$x(t) =x_0 +\int_0^t f(t’)dt’$$
From a conceptual standpoint, you are calculating the total area below the curve $f(t)$ between the point where time is zero and some point $t$, adding the value of the initial position, and jotting down that value as the position of the system at time $t$. Doing this for all points in time give you the position curve. (Remember to count the areas below the speed equals zero axis as negative!)