Displacement As Function Of Time and Periodic Function

Let the initial position of a particle be x0 and position at time t be x. Then its displacement relative to x0 is x – x0 and it depends in some manner on time t. We say that displacement varies with time or is a function of time t.

 We denote the function by ft. Thus x – x0 = ft ,

Here the expression for function depends upon type of motion. If the motion is with uniform velocity we have x – x 0 = Vt linear function Here we say that displacement x – x0 is proportional to t. Position x = x0 + Vt is said to be a linear function of t as power oft is 1. The graph of xt is a straight line. 

If the motion is along the x-axis with uniform acceleration, the displacement is given by x – x0 = ut + ½ 2  at 2. 

Here we have quadratic function oft as the highest power of is 2. The graph of xt is a parabola. If acceleration is not uniform we have an infinite number of ways in which acceleration can change. Of these, one special case is very important-arising from a periodic function of time.

Derivation of Displacement as a Function of Time

In order to understand displacement as a function of time, we will have to derive an expression for displacement also known as the Second equation of motion.

 Let us assume a body traveling with  an initial velocity of v1 at time t1, and is subjected to some constant accelerations thus making its final velocity of v2 at time t2, 

Keeping these things in assumption let’s derive the following.

We know that average velocity is equal to total displacement covered in a given time interval. Thus using this we can say that

V average = Total displacement / Total time  

Displacement = V average Δt

Where Δt  is the change in time and is equal to t2-t1

Since acceleration is constant thus average velocity is mean of initial and final velocity

Therefore ,displacement  = (V1+ V2/2) Δt , where V1 and V2 are initial and final velocity 

Now, since acceleration is constant, final velocity

 V2 = V1 +at 

d = ((V1+V1 + aΔt) /2 ) Δt

 Now we can if we rewrite the above as,

d = (2V₁+ aΔt) Δt /2

The above expression is second equation of motion and is one of the most fundamental expressions in kinematics and is finally reduced to 

d= V1 t+½ at2

Where V1 is the initial velocity and t is the change in time, all the quantities in this derivation like Velocity, displacement and acceleration are vector quantities. 

Thus, the above expression clearly proves that displacement depends upon the time .

Example of an Oscillating Pendulum

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When Bob reaches the highest point, the potential energy is maximum and the kinetic energy is minimum as the velocity is equal to zero but the total energy is conserved throughout the motion and only transformation from kinetic energy to potential energy or vice versa will take place. 

Thus we can easily deduce that the velocity is equal to zero, by looking at the slope of the displacement-time graph at specific times.

The Slope of the graph at A is positive stating that the velocity of the body is also positive, or in the forward, direction while Slope at B is equal to zero meaning that the velocity of the body is zero, while acceleration is still there that later imparts velocity in reverse direction.Thus slope at C is negative in the graph means that the velocity of the body is also negative, or in the reverse direction.

If we draw a displacement-time graph of this oscillating pendulum we will get something like this as shown below in the figure. Here the magnitude of velocity is always positive, it is the direction that decides if the velocity is positive or negative.

[Image to be added Soon]

The displacement of the bob of the pendulum is periodic in nature as it repeats itself after a certain amount of time. Also the displacement at any given time in the future can be predicted with the help of graph since we know the time and the time period of the pendulum. 

Thus we can say that displacement of the oscillating pendulum bob is a function of time.