Formula of Speed Time and Travelled Distance

‘Speed, distance, time’ is one of the famous and important topics within the maths or quantitative field of any competitive exam. The concept of velocity, time, and distance is used drastically for questions referring to specific topics inclusive of motion in a straight line, circular motion, boats, races, clocks, and many others. 

Distance, Speed, Time Formula

Students are commonly asked to determine the distance, speed, or travel time of something given any two variables. These types of problems are quite interesting to solve as it describes real-life situations for many people. For example, a question might say: 

Find the distance a car has traveled in twenty minutes at a constant speed of 50 km /hr. Generally in these problems, we use the distance speed time formula to calculate the desired quantity.

Speed 

Speed is defined as the rate at which an object moves from one place to another in a given interval of time. It is a scalar quantity as it defines only the magnitude not the directions of an object moving. The S.I. unit of speed is m/s.

The speed of a moving object can be calculated as: 

\[ Speed = \frac{Distance}{Time}\]

Speed can either be uniform or variable.

Average Speed: The average speed is the total distance covered by an object in a particular interval of time. For example,

If a moving object covers d₁, d₂, d₃...dₙ with different speeds V₁, V₂, V₃,...Vn m/s in time t₁, t₂, t₃,...tn respectively, the average speed is calculated as:

\[ \frac{Total \, Distance \, Traveled}{Total \, Time \, Taken} = \frac{d_{1},d_{2},d_{3},..d_{n}}{t_{1},t_{2},t_{3},..t_{n}}\]

What is Relative Speed?

Relative speed is the speed of a moving object in terms of another. When two objects are moving in the same direction, then the difference in their speed is termed relative speed.

Similarly, when two objects are moving in different directions, then the sum of their speed is termed relative speed.

Relative Speed Formula in Time and Distance

Let us understand the relative speed formula in time and distance with an example.

If two objects are moving in the same direction at x₁ m/s and x₂ m/s, respectively, where x₁ > x₂, then their relative speed is (x₁ - x₂) m/s.

Example 1: Consider two objects X and Y separated by a distance of d meters. Suppose, If both X and Y are moving in the same direction at the same time at a speed of x meter per second and y meter per second respectively, then

Relative speed = (X - Y) metre per second

If two objects are moving in different directions at x₁ m/s and x₂ m/s, respectively, where x₁ < x₂, then their relative speed is (x₁ + x₂) m/s.

Example 2: Consider two objects X and Y separated by a distance of d meters. Suppose, If both X and Y are moving in different directions at the same time such X moves towards Y at speed of x m/s, and Y moves away from X at a speed of y m/s, where X > Y, then, 

Relative Speed = (X + Y) metre per second

Distance

Distance refers to the length of the path covered by an object or person. You can calculate the distance traveled by an object if you know how long and how fast it moved. The distance traveled by an object or person in terms of speed and time can be calculated as: 

\[ Speed = \frac{Distance}{Time}\]

Time

Time refers to the duration in hours, minutes, or seconds spent to cover a particular distance. Time taken by a moving object to cover a certain distance at a given speed is calculated as :

\[ Time = \frac{Distance}{Speed}\]

Relationship between Speed, Time and Distance

• \[ Speed = \frac{Distance}{Time}\] This tells us how slow or fast an item actions. It describes the distance traveled divided by the point taken to cover the distance. 

• Distance is directly proportional to speed, but time is inversely proportionate. Thus, \[ Distance = Speed * Time \]

• \[ Time = \frac{Distance}{Speed}\], as the rate increases the time taken will decrease and vice versa. 

Using the precise units is critical to not forget while the usage of the formulation. 

Units for Speed, Time, and Distance

Speed, distance, and time may be expressed in one of a kind:

• Time - second (sec), minute (min), hour (hr)

• Distance - meter (m), kilometer (km), mile, feet

• Speed - m/s, km/hr

So, if distance = kilometre and time = hour, then velocity = distance/ time; the units of speed would be km/ hr.

Units of speed, time, and distance are obvious, let us apprehend the conversions related to these.

Speed, Time and Distance Conversions

• In order to convert from km/hour to m/sec, we multiply by 5/18. So, 1 km/hour = 5/18 m/sec

• In order to convert from m/sec to km/hour, we multiply with the aid of 18/five. So, 1 m/sec = 18/5 km/hour = 3.6 km/hour

• 1 yard = 3 ft

• 1 kilometer= 1000 meters

• 1 mile= 1.6 km

• 1 hr = 60 minute = 3600 seconds

If the ratio of speeds is a: b for a certain distance, the ratio of time taken to close the gap may be b: a, and vice versa.

Uses of Speed, Time, and Distance

1. Average speed 

\[common velocity = \frac{general \, distance \, traveled}{overall \, time \, taken}\]

When the distance is consistent: common \[ velocity = \frac{2xy}{x+y}\]; where x and y are the 2 speeds at which the identical distance has been covered.

Solved Examples

1. A person has covered a distance of 60 km in 2 hours. Calculate the speed of the bike.

Solution:

Given: Distance Covered, distance = 60 km,

Time taken, time = 2 hours

Speed is calculated using the formula: \[ Speed = \frac{Distance}{Time}\]

= 60/2

= 30 Km/hr.

2. In a bike race, a biker is moving at a speed of 80 km/hr. He has to cover a distance of 105 km. Calculate the time will he need to reach his destiny.

Solution:

Given: Speed = 80 km/hr,

Distance  to cover, d = 105 km,

Taken time, t =?

Speed is given by the formula: \[ Time = \frac{Distance}{Speed}\] 

Time is taken \[ Time = \frac{Distance}{Speed}\]

= 105/80

= 1.31 hr

Time is taken by the biker = 1.31 hr

3. A car person travels a car at a speed of 50 km per hour. How far can he cover in 2.5 hours?

Solution:

The equation for calculating distance traveled by car, given speed and time, is 

Distance = Speed/Time

Substituting the values, we get

D = 50/2.5

D = 125 km

Hence, a car can travel 125 km in 2.5 hours.

4. If a boy travels at a speed of 40 miles per hour. At the same speed, how long will he take to cover the distance of 160 miles?

Solution:

The formula to calculate time, when speed and distance are given is:

\[ Time = \frac{Distance}{Speed}\]

Time taken by car to cover 160 miles is :

\[ Time = \frac{160}{40}\]

T = 4 hours

Hence, a boy will take 4 hours to cover a distance of 160 miles at a speed of 40 miles per hour.

5. Two boys are running from the same place at a speed of 7 km/ hr and 5 km/hr. Find the distances between them after 20 minutes respectively if they move in the same direction.

Solution:

When boys run in the same direction, 

Their relative speed = ( 7 - 5) km/ hr = 2 km/ hr.

Time is taken by boys = 20 minutes

Distance covered = Speed × Time

\[ = 20 * \frac{20}{60}\]

=6.66

Hence D = 6.6 km

Conclusion

From this discussion, we have concluded that,

• If two moving bodies are moving at the same speed, the distance traveled by them is directly proportional to the time of travel i.e when speed is constant.

• If two moving bodies move for the same time, the distance traveled by them is directly proportional to the time of travel i.e when time is constant.

• If two moving bodies are moving at the same distance, their travel of time is inversely proportional to speed i.e when the distance is constant.