CONSTANT SPEED

Constant speed is also called as uniform rate which involves something travelling at fixed and steady pace or else moving at some average speed. 

For example, A car travels 3 hours. It travels 30 miles in the first hour, 45 miles in the second hour and 75 miles in the third hour.

speed in the first hour = 30 miles/hour

speed in the second hour = 45 miles/hour

speed in the third hour = 75 miles/hour

We have three different speeds in  the three hour journey. 

If we want to find the constant speed for the whole journey of three hours, we have to find the ratio between the total distance covered and total time taken.

That is, constant speed = (30 + 45 + 75)/3

= 150/3

= 50 miles/hr

Based on the above example, the formula is to find the constant speed is given below. 

If a person travels from A to B at some speed, say 'x' miles per hour. He comes back from B to A at different speed, say 'y' miles per hour. Both the ways, he covers the same distance, but at different speeds.

Then, the formula is to find the constant speed for the whole journey is given below. 

Example 1 : 

David drove for 3 hours at a rate of 50 miles per hour, for 2 hours at 60 miles per hour and for 4 hours at a rate of 70 miles per hour. What was his constant-speed for the whole journey ?

Solution :

Step 1 :

Formula for constant speed : 

= Total distance/Total time taken

And also, for distance = Rate ⋅ Time

Step 2 :

Distance covered in the first 3 hours :  

= 50 ⋅ 3

= 150 miles

Distance covered in the next 2 hours : 

= 60 ⋅ 2

= 120 miles

Distance covered in the last 4 hours :

= 70 ⋅ 5

= 350 miles

Step 3 :

Then, total distance is 

= 150 + 120 + 350

= 620 miles

Total time is

= 3 + 2 + 5

= 10 hours

Step 4 :

So, constant speed is 

= 620/10

= 62 miles per hour

Example 2 : 

A person travels from Newyork to Washington at the rate of 45 miles per hour and comes backs to the Newyork at the rate of 55 miles per hour. What is the constant speed for the whole journey ? 

Answer :

Step 1 :

Here, both the ways, he covers the same distance.

Then, the formula to find average speed is

= 2xy/(x + y)

Step 2 :

x ----> Rate at which he travels from Newyork to Washington 

x = 45

y ----> Rate at which he travels from Newyork to Washington 

y = 55

Step 3 :

So, the average speed is

= (2 ⋅ 45 ⋅ 55)/(45 + 55)

= 4950/100

= 49.5

So, the constant speed for the whole journey is 45 miles per hour. 

Example 3 : 

A man takes 10 hours to go to a place and come back by walking both the ways. He could have gained 2 hours by riding both the ways. The distance covered in the whole journey is 18 miles. Find the constant speed for the whole journey if he goes by walking and comes back by riding.     

Solution :

Step 1 :

Given : A man takes 10 hours to go to a place and come back by walking both the ways.

Walking + Walking = 10 hours

2 ⋅ Walking = 10 hours

Walking = 5 hours

Given : He could have gained 2 hours by riding both the ways. 

Riding + Riding = 8 hours

2 ⋅ Riding = 8 hours

Riding = 4 hours

Step 2 :

If he goes by walking and comes back by riding, time taken by him :

Walking + Riding = 5 + 4

Walking + Riding = 9 hours

Step 3 :

Total time taken = 9 hours

Total distance covered = 18 miles

Step 4 :

So, the average speed is

= Total distance/Total time

= 18/9

= 2

So, the required constant speed is 2 miles per hour. 

Example 4 : 

Lily takes 3 hours to travel from place A to place B at the rate of 60 miles per hour. She takes 2 hours to travel from place B to C with 50% increased speed. Find the constant-speed from place A to C.  

Solution :

Step 1 :

Speed (from A to B) = 60 miles/hour

Speed (from B to C) = 90 miles/hour (50% increased)

Step 2 :

Formula to find distance is 

= Rate ⋅ Time 

Distance from A to B is

= 60 ⋅ 3

= 180 miles

Distance from B to C

= 90 ⋅ 2

= 180 miles

Total distance traveled  from A to B is

= 180 + 180

= 360 miles

Total time taken from A to B is

= 3 + 2

= 5 hours

Step 3 :

Formula to find average speed is

= Total distance/Total time 

= 360/5

= 72

So, the constant speed from place A to B is 72 miles/hour. 

Example 5 :

A person takes 5 hours to travel from place A to place B at the rate of 40 miles per hour. He comes back from place B to place A with 25% increased speed. Find the constant speed for the whole journey.  

Solution :

Step 1 :

Speed (from A to B) = 40 miles/hour

Speed (from B to A) = 50 miles/hour  (25% increased)

Step 2 :

The distance traveled in both the ways (A to B and B to A) is same.

So, the formula to find average distance is

= 2xy/(x + y)

Step 3 :

x ----> Speed from place A to B

x = 40

y ----> Speed from place B to A

y = 50

Step 4 :

Average speed = (2 ⋅ 40 ⋅ 50) / (40 + 50)

= 44.44

So, the constant speed for the whole journey is about 44.44 miles/hour. 

Example 6 : 

Distance from A to B = 200 miles

Distance from B to C = 300 miles

Distance from C to D = 540 miles

The speed from B to C is 50% more than A to B. The speed from C to D is 50% more than B to C. If the speed from A to B is 40 miles per hour,  find the constant speed from A to D. 

Solution :

Step 1 :

Speed (from A to B) = 40 miles/hour

Speed (from B to C) = 60 miles/hour (50% more)

Speed (from C to D) = 90 miles/hour (50% more)

Step 2 :

Formula to find time is 

= Distance/Time 

Time (A to B) = 200/40 = 5 hours

Time (B to C) = 300/60 = 5 hours

Time (C to D) = 540/90 = 6 hours

Total time taken from A to D is

= 5 + 5 + 6

= 16 hours

Total distance from A to D is 

= 200 + 300 + 540

= 1040 miles

Step 3 :

Formula to find average speed is

= Total distance/Total time

= 1040/16

= 65

So, the constant speed from A to D is 65 miles per hour. 

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