Displacement as a Function of Time

To explain displacement as a function of time, we must first derive a displacement expression, commonly known as the second equation of motion.

Consider a body moving at v1 at time t1, subjected to constant accelerations, resulting in v2 at time t2.

The key assumptions

Assuming these assumptions, let’s come up with the following.

The entire distance travelled at a particular time equals the average velocity. That’s why V average is defined as total displacement and total time.

V average Δt is the displacement formula.

ΔT is the rate of change, and it is equal to the product of two previous times, t2 and t1. It follows that average velocity is the sum of starting and final velocity because acceleration is constant.

It means that the displacement equals the sum of two velocities: the initial and end velocity. i.e., = (V1 + V2/2) Δt.

Final velocity may be calculated because of the constant acceleration.

V2 is =V1 + at.

In other words, d = ((V1 + V1 + aΔt /2) (t)

Now that we’ve rewritten the preceding text,

In other words, d = (2V1 + a ΔT)Δt/2

As the second equation of motion, this expression is a foundational one in kinematics. The formula can ascertain it:

d= V1 t + ½ at2

All the numbers in this derivation, such as velocity, displacement, and acceleration, are vector quantities, where V1 is the initial velocity and t is the time change. Thus, the following statement demonstrates that displacement is a function of time.

Oscillating pendulums example

• Even though Bob’s kinetic and potential energy is at their lowest points as he nears the top of the trajectory, the total amount of energy remains constant during his journey. Only the conversion from kinetic to potential energy or the other way around occurs.

• As a result, looking at the slope of the displacement-time graph at specified points allows us to determine that the velocity is zero.

• As a result, the slope at A is positive, indicating that the body’s velocity is positive. In contrast, the slope at B equals zero, showing that the body’s velocity is zero, yet acceleration continues to impart velocity in the opposite direction.

• In other words, if the slope at point C on the graph is negative, then the body’s velocity is negative as well.

• It is what we’ll get if we plot the displacement vs time for this oscillating pendulum, as shown in the picture below, rather than relying on the magnitude to determine whether or not the velocity is positive or negative. In this case, the direction is used instead.

• As the bob of the pendulum swings back and forth, the movement is cyclical. 

• Since we know the time and period of the pendulum, we can also anticipate the displacement at any point in the future using a graph.

• Thus, the displacement of the oscillating pendulum bob can be described as a function of time.

The graph of motion

Now, if we plot the bob’s displacement with time, we’ll get something like this:

Displacement of function as a time of periodic motion

Let x0 be the particle’s starting position, and x t be its current position. In this case, its distance from x0 is equal to the difference between x and x0. Time t influences this in some way. When a location change occurs over time, we say that the change is a function of time t. The function is referred to as f(t). 

x – x0 = Vt if the motion is uniformly accelerating (linear function). Displacement x – x0 is directly proportional to t, as shown in this figure. As long as the power of t is 1, the position x is considered a linear function of t. x(t) is a straight line on the graph. Assuming a homogeneous acceleration along the x-axis, the displacement can be expressed as 

x – x0 = ut+½ at2.

As the largest power of t is 2, we have a quadratic function of x(t) as a parabola on the graph. Acceleration can fluctuate in unlimited ways if it isn’t uniform. One of these is particularly significant: it arises from the periodicity of time.

Velocity-time

• The relationship between velocity and time is straightforward during uniformly accelerated, straight-line motion. It takes longer for the acceleration to have a significant effect on the speed. With constant acceleration, the rate of change in velocity is a function of time. 

• Speed should rise by twice the amount in twice the time if it increases by a given amount in a certain length of time. The new velocity equals the old velocity plus the change in initial velocity. You should already be able to visualise the equation in your head.

Algebraically, this is the simplest of the three equations to deduce. The definition of acceleration is the starting point.

a= ∆ v/∆t

By expanding ∆v to v-v0 and condensing ∆t to t, we get:

a= v-v0/t

Conclusion

We have thus learned the basic concept of displacement as a function of time. It is a periodic function that means that it can be anticipated at any point in the future, provided that we know the time and have a good understanding of how long the pendulum has been in motion. Hence, it is a function of time to state that displacement is a function of time.