Speed is a measure of how quickly an object moves. It is the **distance** traveled by a body in **unit** time. For example, if a kid runs 50 meters in 10 seconds, we can say that he moves 5 meters every second. The unit time in this example is one second. So his speed is 5 meters per second.

The following formula relates speed to distance and time.

**✩** Speed Formula

$Speed=TimeDistance $

where:

- Distance = Distance traveled by the moving object
- Time = Time taken to travel the distance

We use this formula to calculate the speed of a moving object.

The same formula can be used to calculate distance or time:

- Distance = Speed × Time
- $Time=SpeedDistance $

It is not necessary to remember these as the original formula is sufficient for calculating distance or time.

### 1 Example

A car goes a distance of 200 miles in 2 hours. What is its speed?

#### Solution

Applying the speed formula:

s = d/t

s = 200/2 miles/hour

s = 100 miles/hour

Speed of car = 100 miles/hour.

### 2 Example

The distance between the two cities is 200 miles. A bus takes 4 hours to cover this distance. Calculate the speed of the bus?

#### Solution

Distance = d = 200 miles.

Time = t = 4 hours.

Speed = s = unknown

Using the speed formula:

s = d/t

s = 200/4 = 50 miles / hour.

Speed of bus = 50 miles/hour.

## Unit of Speed

Several units of speed are possible. They depend on the units used to measure distance and time. Some examples:

- Car: 80
**miles/hour** - Ship: 20
**nautical miles/hour** - Rocket: 500
**miles/minute** - Train: 100
**kilometers/hour** - Athlete: 8
**meters/second**

It is possible to convert one unit to another. Let us see some examples.

### 3 Example

A train travels at 108 Km/hour. Convert its speed to meters/second.

#### Solution

1 hour = 60 minutes = 60×60 seconds = 3600 seconds

1 Km = 1000 meters

Speed = 108 km/hour

$=1hour108km $

$=3600seconds108×1000meters $

= 1080/36 meters/seconds

= 30 meters/second

### 4 Example

A rocket travels 500 miles per minute. What is its speed in miles per hour?

#### Solution

1 minute = 1/60 hours

Speed = 500 miles/minute

$=1minute500miles $

$=601 500 hourmiles $

= 500 × (60/1) miles/hour

= 30000 miles/hour

## Calculating Speed, Distance, and Time

We can use the speed formula to calculate distance and time also. However, before applying it, we have to make sure that units of the known quantities are consistent.

### Distance Calculation

Distance = Speed × Time

Make sure that the units of speed and time are consistent. For example, if speed is in miles per **hour** and time **minutes**, we have to do **one** of the following conversions:

- Time to hours
- Speed to miles/minute

### Time Calculation

$Time=SpeedDistance $

Make sure that the units of speed and distance are consistent. For example, if speed is in **kilometers** per hour, and distance in **meters**, we have to do **one** of the following conversions:

- Distance to kilometers
- Speed to meters/hour

**✩** Consistent Units are Must!

- Check that the units are consistent before using the speed formula for distance or time calculation.
- Convert units if necessary.

### 5 Example

A cyclist completes a race in 150 minutes at a speed of 16 miles per hour. How long is the race?

#### Solution

Distance = d = unknown

Time = t = 150 minutes

Speed = s = 16 miles / hour

The speed is given in miles per hour, whereas the time is in minutes. Before applying the speed formula, we have to convert either the time to hours or the speed to miles per minute. It is easier to convert the time, so let us do that.

Time = t = 150 minutes

$=60150 $ hours (There are 60 minutes in one hour)

= 15/6 hours

t = 5/2 hours

Now we can apply the formula:

s = d/t

$16=(5/2)d $

16 = d × (2/5)

80 = 2d

d = 40 miles

Distance covered by cyclist = 40 miles

## Average Speed

While going from one point to another, an object may travel at different speeds. Its average speed is the ratio of total distance and total time elapsed during the journey.

$AverageSpeed=TotalTimeTotalDistance $

Let us understand with an example.

### 6 Example

A train goes from station A to station B at a speed of 200 Km/h. While returning, the train has a better engine. It is faster by 100 Km/h than the old engine. What is the train’s average speed for the round trip?

#### Solution

Let the distance between stations = d Km.

Speed from A to B = 200 Km/h

Speed from B to A = 200 Km/h + 100 Km/h = 300 Km/h

From the speed formula:

Time = Distance/Speed

Time from A to B = t_{AB} = d/200

Time from B to A =t_{BA} = d/300

Total Time = t_{AB} + t_{BA} = d/200 + d/300

$=200×300300d+200d =200×300500d $ hours

Let us calculate the average speed now.

Total distance = AB + BA = d + d = 2d

AverageSpeed = TotalDistance/TotalTime

$=200×300500d 2d $

$=500d2d×200×300 $

$=5400×3 =240$ Km/h

Average Speed = 240 Km/h

## Questions

These questions cover speed and unit conversion concepts discussed above. Some of these are challenging. All the best!

### 1 Question

A ship goes from one port to another in 2 days and 4 hours. Its average speed is 20 miles/hour. How far are the ports?

#### Answer

Can you convert time to hours and apply the speed formula?

### 2 Question

A drone travels 5 miles to deliver a package. Its speed is 40 miles/hour. How much time does it take to reach its destination?

#### Answer

Can you apply the speed (= distance/time) formula?

### 3 Question

An athlete runs 5 meters per second. What is her speed in kilometers per hour?

#### Answer

Can you use 1 second = 1/3600 Hour and 1 meter = 1/1000 Kilometer for conversion?

### 4 Question

A ship goes 20 nautical miles in an hour. Calculate its speed in meters per second?

(1 nautical mile = 1.85 Km)

#### Answer

Can you use 1 hour = 3600 seconds and one nautical mile = 1850 meters for conversion?

### 5 Question

A bus left town A for town B. Having traveled 300 km, it stopped for 30 minutes due to road blockage. It traveled 60% of the total distance by this time. After it started again, the driver increased the speed by 20 km/h and reached town B at the scheduled time. What was the original speed of the bus?

#### Answer

Can you find the relation between the original speed and time till bus hits the road block?

## Answers

### 1 Answer

Distance = d = unknown

Time = t = 2 days and 4 hours

Speed = s = 20 miles/hour

The unit speed is in miles per hour, while time is in days. To make units consistent, we convert time to hours before applying the formula.

Time = t = 2 days and 4 hours = 2 × 24 + 4 = 48 + 4 = 52 hours.

Now we can apply the speed formula:

s = d/t

20 = d/52

20 × 52 = d

d = 1040 miles

Distance between ports = 1040 miles

### 2 Answer

Distance = d = 5 miles

Time = t = unknown

Speed = s = 40 miles / hour

The units are consistent, so we can directly apply the formula.

Using the speed formula:

s = d/t

40 = 5/t

t = 5/40 = 1/8 hours

Time taken by drone = 1/8 hours

### 3 Answer

1 hour = 60 minutes = 60×60 seconds = 3600 seconds.

Therefore 1 second = 1/3600 hour

1 Kilometer = 1000 meter

So 1 meter = 1/1000 kilometer

Given Speed = 5 meters/second

$=1second5meters $

$=(1/3600)hour5×(1/1000)kilometers $

Converting fractional division to multiplication by inverting the denominator:

= (5/1000) × (3600/1) kilometers/hour

$=1000×15×3600 $ kilometers/hour

= 180/10 kilometers/hour

= 18 km/hour

### 4 Answer

1 hour = 60 minutes = 60×60 seconds = 3600 seconds.

1 nautical mile = 1.85 kilometer = 1.85×1000 = 1850 meter

Ship Speed = 20 nautical mile/hour

$=1hour20nauticalmile $

$=1×3600seconds20×1850meters $

= (2 × 185)/(36) meters/second

= 10.27 meters/second

### 5 Answer

The bus stops at point C for 1/2 hour due to blockage.

Let the initial speed be x Km/h

Time = Distance/Speed

Time to travel the distance AC = t_{AC} = 300/x hours

Let’s first calculate CB

60% of AB = 300 Km

AB × 0.60 = 300

AB = 300/0.60 = 500 Km

CB = Remaining distance = 500 − 300 = 200 Km

Speed of bus after point C = (x + 20) Km / h

Time to travel the distance CB = t_{CB} = 200/(x + 20) hours

Had there been no blockage, the bus would have traveled at xKm/h. Its travel time would have been:

t_{usual} = AB/x = 500/x hours

The bus reached town B on time, despite blockage, therefore:

Usual time = Time to travel the distance AC + Delay + Time to travel the distance CB

$t_{usual}=t_{AC}+21 +t_{CB}$

$x500 =x300 +21 +x+20200 $

Multiplying both sides by 2x(x + 20):

500 × 2(x + 20) = 300 × 2(x + 20) + x(x + 20) + 200 × 2x

1000x + 20000 = 600x + 12000 + x^{2} + 20x + 400x

1000x + 20000 = x^{2} + 1020x + 12000

0 = x^{2} + 1020x − 1000x + 12000 − 20000

x^{2} + 20x − 8000 = 0

The speed is calculated by solving the above quadratic equation. Factorizing the quadratic expression we get:

(x + 100)(x − 80) = 0

x = − 100, or x = 80.

Speed is not a negative quantity, so x = 80Km/h.

## Related

**Challenging Speed, Time, Distance Questions (Hints & Answers) ➤**