Analyzing Motion: Deriving Displacement Graphs from First Principles

• Old_sm0key
In summary, the conversation is about deriving the displacement and velocity graphs of a falling object from first principles, using initial conditions and equations of motion. It is suggested to use integrals to find the equations for different intervals of time, but there is a concern about the velocity discontinuity at the bounce. The conversation also touches on the use of a spring model to smooth out the curve and finding the right equation for the displacement graph. Different methods for finding the equation are discussed, including using indefinite or definite integrals, and the use of mod() in spreadsheets.
Old_sm0key
Homework Statement
Not a homework question per se. I revisited introductory kinematics, as taught at British A Level, and am struggling to reproduce the stadnard plots for displacement and velocity versus time.
Relevant Equations
The standard 1D kinematics equations plus introductory differentiation
Initial displacement is h above the ground ie ##s\left ( t =0\right )=h##. I've chosen the ground as the vertical origin with upwards as the positive direction. Gravity will therefore always act in negative direction throughout. Here are the graphs I which to reproduce from first principles, where the motion starts at the red vertical line owing to my initial conditions:

##0\leqslant t< A##
Starting from ##v=u+at## we derive displacement thus:
$$\int_{0}^{t} vdt^\prime=\int_{0}^{t}\left ( u+at \right )dt^\prime$$
$$s\left ( t \right )-h=ut+\frac{1}{2}at^2$$
Using conditions ##t=0,u=0, a=-g\, \forall t \to s\left ( t \right )=h-\frac{1}{2}gt^2## which would give a parabola for first portion of displacement graph. So happy.
The corresponding velocity graph is ##\frac{\mathrm{d} s}{\mathrm{d} t}=-gt##, so giving a negative straight line. Happy.

##A\leqslant t< B##
Now I'm stuck with the displacement curve because I cannot exploit ##t=0## on the integral limits to simplify the maths...?
Plus it seems a fudge to suddenly invoke ##s\left ( t \right )=vt-\frac{1}{2}at^2## that I see in some textbooks.

If you want to avoid the fudge of the discontinuity of the velocity at the bounce, you can model the ball as having a spring attached to its bottom. This will round off the curve with a sinusoidal and give it a collision duration of half a period. You can sharpen it by increasing the spring constant ##k.## so that the period of oscillations is much much smaller than the time between bounces.

With what are you trying to reproduce them? If in a spreadsheet, it's just a matter of first converting t to be in the range 0 to A using MOD.

Thanks both, however I probs didn't make myself too clear.

(I accept the velocity discontinuity @ bounce ie precluding differentiation)

I want to derive the s vs t equation, just like I did above, now for ##A\leqslant t< B## to produce the shape shown in plot. Just can't get the right s(t) function..!

Old_sm0key said:
$$\int_{0}^{t} vdt^\prime=\int_{0}^{t}\left ( u+at \right )dt^\prime$$
In order to correctly distinguish between your integration variable and the limits, this should be$$\int_{0}^{t} vdt^\prime=\int_{0}^{t}\left ( u+at^\prime \right )dt^\prime$$If I’ve understood what you are asking…

Let T be the duration of each ‘cycle’ (from t=0 to A). This is a constant you can express in terms of h and g.

You have chosen to make B halfway through the 2nd cycle. Not sure if that’s deliberate, but no matter.

You can use the method you used for the 1st cycle - but change the integration limits: from T to t+T/2. Because the process is cyclic, your initial conditions at the start of each cycle are identical to those at t=0.

Old_sm0key said:
Thanks both, however I probs didn't make myself too clear.

(I accept the velocity discontinuity @ bounce ie precluding differentiation)

I want to derive the s vs t equation, just like I did above, now for ##A\leqslant t< B## to produce the shape shown in plot. Just can't get the right s(t) function..!
Why not include B in the interval? I.e. ##\ A\le t \le B \ ##
You can actually go father, to the next zero of ##s(t)##, maybe call it ##C##.

At ##t=B## you have ##s(t) = h## and ##v(t) = 0##.

Either use indefinite integrals and evaluate the constants of integration according to the above values (at ##t=B## ) ,

or use definite integrals and integrate from ##t## to ##B##, or from ##B## to ##t##. Just be consistent with the direction of the integration.

Old_sm0key said:
derive the s vs t equation
In post #1 you said you wanted to plot the function, and I gave you a way to do that in post #3. Now you say "derive" it, but in what sense do that? From a differential equation, or as an integral?

Or are you just looking to write the function s(t)? Depends what functions you accept as valid in it. There will have to be one which is not continuously differentiable. As I mentioned, there is mod(), which is not only used in spreadsheets.
https://en.wikipedia.org/wiki/Modulo_operation

1. What is "Analyzing Motion"?

"Analyzing Motion" is a scientific process that involves studying the movement of objects and analyzing the data collected to understand the underlying principles and patterns of motion.

2. What is the importance of deriving displacement graphs from first principles?

Deriving displacement graphs from first principles allows us to accurately represent an object's position over time and determine its velocity and acceleration. This information is crucial in understanding the motion of an object and can be used to make predictions and calculations.

3. How do you derive displacement graphs from first principles?

To derive a displacement graph from first principles, you must first collect data on an object's position at different points in time. Then, you can use this data to calculate the object's velocity and acceleration, which can be plotted on a graph to show the changes in displacement over time.

4. What are some common mistakes when analyzing motion and deriving displacement graphs?

Some common mistakes when analyzing motion and deriving displacement graphs include not collecting enough data points, not considering external factors that may affect the motion of the object, and incorrect calculations. It is important to carefully collect and analyze data to ensure accurate results.

5. How can analyzing motion and deriving displacement graphs be applied in real-world situations?

Analyzing motion and deriving displacement graphs can be applied in various fields, such as physics, engineering, and sports. In physics, it can help us understand the laws of motion and make predictions about the behavior of objects. In engineering, it can be used to design efficient and safe structures. In sports, it can help athletes improve their performance by analyzing their movements and identifying areas for improvement.

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