## What are time and time-related variables?

Time is the event progression starting from past to present and then, future statistically. The time-related variable is any particular value in relation to time to assess its changes on a specified (XY) plane.

It is used in potential and kinetic energy, Mechanics, Thermodynamics, equations related to aerospace, etc.

## Dimensional analysis of time

Time is expressed in seconds ‘s’ and its dimensional formula is [T]. In science, technology, engineering and maths (STEM) give us a method to analyze the relationship of various physical, and fundamental quantities. [M], [T], and [L] known as mass, time and length respectively are fundamental quantities and dimensions of the above mentioned. For example, velocity is represented as [LT^{-1}] and force as [MLT^{-2}].

## Environmental engineering

It deals with the conservation of resources by using renewable sources and researching technical ways to adopt practices for environment-friendly techniques along with cost analysis for prediction of long term expenses and savings.

### Stormwater runoff

To calculate the peak stormwater runoff rate during heavy rainfall, the below formula is required wherein C_{f} is different for different locations.

$Q={C}_{f}CIA$

Here,

Q = peak runoff rate of storm water (ft^{3}/s)

C_{f }= runoff coefficient adjustment factor

C = runoff coefficient

I = rainfall intensity (in./h)

A = drainage area (acres)

Thus, Q is the time-related variable.

### Water supply (Hazen-Williams formula)

It is the mathematical expression for evaluation of head loss. It uses the flow of water and the properties of the pipe and is helpful for irrigation and sprinkler systems.

${h}_{f}=\frac{10.67L{Q}^{1.85}}{{C}^{1.85}{d}^{4.87}}$

Here,

h_{f}= head loss due to friction

L = length of pipe (ft)

Q = water flow rate (gallons/minute)

C = Hazen William’s constant

d = diameter of the pipe (in.)

Thus, ‘Q’ is the time-related variable.

## Soil mechanics

It deals with soil exploration and tests, the concept of properties of various soils, and for analyzing, how can it be beneficial for man-made structures.

### Constant head permeability test

It is the laboratory method used for coarse-grain soil.

$k=\frac{QL}{Ah}$

Here,

Q=Volume of water collected in time t

L=distance between manometer taping point

A=Cross sectional Area of sample

h= difference in manometric liquids

Thus, ‘Q’ is the time-related variable.

### Falling head permeability test

It is the laboratory method used for fine-grain soil.

$K=\frac{2.3aL}{At}{\mathrm{log}}_{10}\left(\frac{{h}_{1}}{{h}_{2}}\right)$

Here,

a = area of a tube in m^{2}

A = area of the sample in m^{2}

t = time in ‘sec’

L = length in ‘m’

h_{1} = upstream level edge at t=0

h_{2} = upstream level after time ‘t’

Thus, ‘h_{1}’ and ‘h_{2}’ are time-related variables.

### Consolidation equation

Consolidation of soil is the process by which the soil changes the volume gradually when there is a pressure change, the water oozes out due to pressure, and thus the soil shrinks (in volume).

k=C_{v}γ_{ω}m_{v}

Here,

γ_{ω}= unit weight of water (N/cm^{3})

C_{v}= coefficient of consolidation (cm^{2}/sec)

m_{v}= Volume compressibility (cm^{2}/N)

Thus, ‘C_{v}’ is the time related-variable.

## Fluid mechanics

It is the study of water or any liquid which flows freely and to analyze its behavior, characteristics, etc.

### Continuity equation (conservation of mass)

When comparing any 2 substances having density, area, and volume, equating both of them refers to the continuity equation.

ρ_{1}A_{1}V_{1}=ρ_{2}A_{2}V_{2}

Here,

- ρ = density
- A = area
- V = volume

*Boyle’s Law: *P_{1}V_{1}=P_{2}V_{2}

*Gay-Lussac’s Law: *$\frac{{P}_{1}}{{T}_{1}}=\frac{{P}_{2}}{{T}_{2}}$

*Charles’ Law: *$\frac{{V}_{1}}{{T}_{1}}=\frac{{V}_{2}}{{T}_{2}}$

Here,

- P= absolute pressure
- F= Force
- A=Area
- V=volume
- T= absolute temperature
- Thus, ‘A’ is the time-related variable.

## Traffic engineering

Comprises of different vehicular-related expressions and how is it affecting the lives of people and analyzes the output.

### Time-mean speed

It is the average speed estimated of all the vehicles for a specific time period.

${V}_{t}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum {V}_{i}}}}}{n}\phantom{\rule{0ex}{0ex}}$

Here,

- V
_{t}= time mean speed (km/hr) - V
_{i}= observed instantaneous speed of i^{th}vehicles (km/hr) - n = number of vehicles observed

hourly expansion factor=$\frac{Totaltrafficvolumerforaday}{Trafficvolumeforparticularhour}$

daily expansion factor=$\frac{Averagetotalvolumeofaweek}{Averagevolumeforparticularday}$

monthly expansion factor=$\frac{AnnualAveragedailytraffic}{Annualdailytrafficofparticularmonth}$

Travel time per unit length of road =$T=\frac{3600}{V}$

Here,

- V = Speed in km/hr
- Thus, ‘V
_{t}’, ‘V_{i}’ and ‘V’ are the time-related variables.

## Time and project cost- independent and dependent variables

As the time for an activity increases, the cost required for it also generally increases. Thus, the cost can be described as a variable that depends on time, which is independent. Or in other words, cost and time can be said to be dependent and independent variables respectively. Thus, for the successful completion of any project, the influence of time on project characteristics needs to be carefully considered. While scheduling any project, the effects of unforeseen complications, risks, and delays also need to be taken into account to avoid inconveniences during the project.

## PERT and CPM

PERT Definition: The abbreviation stands for program evaluation and review technique. It relies on the probabilistic approach and consists of 3 ‘time estimates’ (t_{o}, t_{p}, t_{m}) according to mathematical standards.

Expected completion time of an Activity (t_{E})

${t}_{E}=\frac{{t}_{o}+4{t}_{p}+{t}_{m}}{6}$

Where,

- t
_{o}= Optimistic time, - t
_{p}= Pessimistic time *t*= Most likely time_{m}

The Standard deviation of an activity (σ)

$\sigma =\frac{{t}_{p}-{t}_{0}}{6}$

The Variance of an activity

${\sigma}^{2}={\left(\frac{{t}_{p}-{t}_{o}}{6}\right)}^{2}$

Slack (s): time by which an event may be delayed without affecting completion time.

Thus, ‘t_{o}, t_{p}, t_{m}’ are time-related variables.

CPM Definition: It means Critical Path Method. It relies on the determination approach in which 1 ‘time estimate’ is made.

Activity times are Early start time (EST), Early finish time (EFT) = EST + Activity time, Latest finish time (LFT), and Latest start time (LST) = LFT – Activity time.

Float is the range in which activity or finish time fluctuates without affecting the project completion. Types include total float, free float, independent float, and interfering float.

*Cost slope* = $\frac{Crash\mathrm{cos}t-Normal\mathrm{cos}t}{Normaltime-Crashtime}$

Thus ‘ t_{c}, t_{0pt} and t_{n}’ are time-related variables.

## Time series in data science

A time series is a statistical plot of a quantity vs time. For example, in the case of a share market, we plot the price of the share regarding time. Here, time is an independent variable, while the share price depends on various factors. The purpose of time series is to forecast the future.

In data science, there are two kinds of time series, one is univariate, and another is multivariate. If there is only one variable that is changing with time, then it is univariate times series. For example, when a sensor is measuring room temperature every second, then it works only with one variable, which is temperature. If a sensor is measuring the temperature of two different rooms, then it is working with two variables. If we plot the data points with time, then we will get multivariate time series.

As we know peak electricity consumption is during the daytime, here consumption is one dependent variable and if the power supplier is charging more during peak hours, then the price also becomes a dependent variable, and time is the independent variable. To find out the cost per unit during peak hours, we use a regression model. Regression is the measurement of dependency between two dependent variables.

But if there are no extra charges for peak hours then we have only one dependent variable which is the price/unit consumption. If we want the total expense of electricity, then we just need the units consumed data. This is the deterministic model where there is a direct relationship between two variables.

Autocorrelation is a repetition of similar time series data over a period.

## Seasonality

It refers to periodic fluctuation. For example, electricity consumption is high in summer and the sale of winter clothes is high in winter.

## Important conversions and units

- 1 day = 24 hours
- 1 hour = 60 minutes
- 1 minute = 60 seconds
- 1 kWh = 3,60,00,000 J
- 1N = 1kg m/s
^{2} - 1 W = 1 J/s
- 1 Hz = 1 s
^{-1}

## Context and Applications

Data scientists use time-series data for statistical analysis, various mathematical datasets, machine learning, time-series data prediction, and so on. Time and time-dependent variables are the fundamentals to learn any analytical skills related to economics or engineering. Any engineering discipline will provide the necessary analytical skills to become a data scientist.

- Bachelors in Computer Science
- Bachelors in Science in Statistics
- Bachelors in Science in Mathematics
- Bachelors of Technology in Civil Engineering

## Practice Problems

1. What is the Dimensional Formula of time?

T

(T)

[T]

T

Correct option- c

Explanation: [T] is the Dimensional Formula of time.

2. 1kWh is how many joules?

3.6 x 10

^{7 }J36 x 10

^{7 }J3.6 x 10

^{-7 }J36 x 10

^{-7 }J

Correct option- a

Explanation: 1kWh is 3.6 x 10^{7 }or 3,60,00,000 J.

3. What is ‘Q’ in the Constant Head Permeability Test Equation?

Rate of water collected in time ‘t’

Volume of water collected in time ‘t’

Speed of water collected in time ‘t’

None of these

Correct option- b

Explanation: ‘Q’ is the Volume of water collected in time ‘t’ in the Constant Head Permeability Test Equation.

4. CPM and PERT stand for respectively?

Critical Projection Method and Program Effect and Review Technique respectively

Central Path Method and Program Evaluation and Relation Technique respectively

Central Projection Method and Program Effect and Relation Technique respectively

Critical Path Method and Program Evaluation and Review Technique respectively

Correct option- d

Explanation: CPM and PERT stand for Critical Path Method and Program Evaluation and Review Technique, respectively.

5. If the slack time is greater than zero (s>0), then it is called a?

Super-critical event

Sub-critical event

Critical event

Non-critical event

Correct option- b

Explanation: If the slack time is greater than zero (s>0), then it is called a Sub-critical event.

## Related Concepts

- Velocity
- Frequency
- Oscillations
- Distance
- Speed

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