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Here, we will see all the equations of the position of a particle as a function of time and the explanation of the terms used in it. Before moving ahead, one should know what the functions of time are? Is the position a function of time only or also velocity? Likewise, is velocity a function of time only or also the position? d2x/dt2=−n2x,
where n is a positive constant. This is an example of a second-order differential equation. It can be shown that the general solution to this equation is x(t)=Asinnt+Bcosnt, where A and B are constants.
The following are functions of time:
s(t) = distance a particle travels from time 0 to t.
v(t) = velocity of a particle at time t.
a(t) = acceleration of a particle at time t.
If we want to see how the position of a particle changes concerning time only, then its velocity must remain constant with time. Likewise, if we want to see how velocity varies with time, then the distance between the former position of the particle and the current position should remain constant with time. Similarly, if we want to see how acceleration varies with time, then the difference between the initial velocity U and final velocity V should remain constant. Is this what the above functions of time tell us?
We want to give another example: p(y) = water pressure at depth y below the surface. Water pressure is given by: p=ρgh. Here the density ρ has to be constant if pressure is only the function of depth y.
Position Of a Particle as a Function of Time
The position of a particle moving in a straight line is a vector that represents a point P on the line in relation to the origin O. The position of a particle is often thought of as a function of time, and we write x(t) for the position of the particle at time t.
The displacement of a particle moving in a straight line is a vector defined as the change in its position. If the particle moves from position x(t1) to position x(t2), its displacement is x(t2)−x(t1) for the time interval [t1,t2].
The distance travelled by a particle is the ‘actual distance’ travelled.
Equation Of Position Of Particles as a Function of Time
The key notions used to describe the motion of objects are:
- t: the time, measured in seconds [s].
- X (t): the position of an object as a function of time—also known as the equation of motion. The position of an object is measured in metres [m].
- V (t): the object’s velocity as a function of time. Measured in [m/s].
- a(t): the acceleration of the object as a function of time. Measured in [m/s2].
- xi=x(0),vi=v(0): the initial (at t=0) position and velocity of the object (initial conditions).
Three functions characterise the motion of an object: the position function x(t), the velocity function v(t), and the acceleration function a(t). The functions x(t), v(t), and a(t) are connected—they all describe different aspects of the same motion.
You are already familiar with these notions from your experience driving a car. The equation of motion x(t) describes the car’s position as a function of time. The velocity describes the change in the position of the car, or mathematically,
v(t)= rate of change in x(t).
If we measure x in metres [m] and time t in seconds [s], then the units of v(t) will be metres per second [m/s]. For example, an object moving at a constant speed of 30[m/s] will change its position by 30[m] each second.
The rate of change of the velocity is called the acceleration of a particle.
a(t)≡rate of change in v(t).
Acceleration is measured in metres per second squared [m/s2]. Constant positive acceleration means the velocity of the motion is steadily increasing, like when you press the gas pedal. Constant negative acceleration means the velocity steadily decreases, like when you press the brake pedal.
Conclusion
At the end of the article, readers may know how to use the equation of motion and understand concepts like velocity and acceleration well. You will also learn how to easily recognise which equation is appropriate to use to solve any given physics problem. If we say s(t), I think it implies that everything has to be constant but time. Otherwise, if displacement s is a function of more than functions of time, for example, if it’s a function of ‘time’ and ‘velocity’, we should write s(v,t).
