# How can you determine an object's displacement as a function of time in real life?

Image you have an object, say a projectile or a vehicle, moving with non-uniformly accelerated motion. If you were to measure its position in space say, every second or every tenth of a second, is it possible to construct a displacement vs. time function out of your data, in order to differentiate it and find the velocity vs. time and acceleration vs. time functions?

In physics textbooks, many problems ask you to find an object's velocity or acceleration at a given time, knowing the object's displacement as a function of time, say, $x(t)=12t³+6t.$ Can functions like those be determined in real life at all, or are they purely theoretical?

• Why not? You just need a proper measuring tool. However the function you get would be numerical, not analytic. So you would need to either fit some analytic function to your data or else take your derivatives numerically. – safesphere Nov 11 '17 at 20:44
• Hold a ruler next to the object, and take a video recording as it moves. – Whit3rd Nov 12 '17 at 0:48
• @Whit3rd In what way would that yield a function of position with respect to time? Obviously you can take a series of measurements using a video recording. What I'm asking is how can you construct the MATHEMATICAL EQUATION of the displacement vs. time function, x(t) using the measurements. – JuanEsteban Valdez Nov 12 '17 at 1:09
• You can compare the mathematical equation to measurements. Constructing an equation, however, is NOT a measurement function, but an intellectual one. – Whit3rd Nov 12 '17 at 1:15

Yes you can make the measurements as you described and determine objects position as a function of time. The velocity can be determined from these values as well. If we had no clue to the physics driving the motion, we could still fit the objects position and velocity with curves of some sort. We usually have a clue to the physics driving the motion (as in gravity acting in a certain direction) and often able to fit a physics-based curve to the data. This method is used to track asteroids and comets, for example. We measure the position over a period of time and can then calculate its predicted position at future times.

• What sort of maths does one use to "fit a curve" to a given set of points obtained in measurements, as you describe? – JuanEsteban Valdez Nov 12 '17 at 1:11
• there are routines that allow the user to do a least squares type of fitting to a set of data points. These routines are mostly just algebraic expressions to minimize the errors. Best to look up or google least squares fit. It's a pretty common technique. If you try to fit a straight line (y=ax+b) to a set of points, the least squares fit would determine the value of a and b which give best overall fit. – jmh Nov 12 '17 at 2:25

It depends a lot on the type of object.

If it's a star, we can observe the red or blueshift of its emission to get an estimate of its velocity towards or away from us, and observe it with telescopes to measure its transverse motion.

If it's an electron, you might need to produce multiple subjects moving the same way and measure different ones at different times realtive to some reference event (say, move a phosphor screen closer or nearer to an aperture the particles are flowing through, to see how they move after they exit the aperture)

If it's an airplane you aren't in, you can track it with a radar system.

If it's a car, there are lots of approaches you might take. You can record the readings from its speedometer, or use a GPS system, use a radar system from outside the car, or just track it with a theodelite, to measure its location.

Depending how you measure the object's location, there are going to be more or less (but never no) errors in your measurement. You'll have to work out how to deal with these.

Then you can fit the measurements to a polynomial or any other function you think is a likely candidate to describe the object's motion. If you're tracking an airplane with radar, this is one of the most studied problems in signal processing, and is the reason the Kalman filter was invented (a technique that would apply well to your vehicle tracking problem also).