Time, Speed & Distance

Relation Between Time, Speed, and Distance

Speed = Distance / Time
If time is constant, there is a direct relationship between speed and distance. For example:
- Suppose A and B are traveling with speeds of 60 km/h and 40 km/h respectively, and the time t is fixed at 2 hours.
- The distance traveled by A = 2 * 60 = 120 km
- The distance traveled by B = 2 * 40 = 80 km
- The ratio of their speeds is 60:40 or 3:2.
- The ratio of their distances is 120:80 or 3:2, showing that distance is directly proportional to speed when time is constant.
If the distance is constant, speed and time are inversely proportional to each other. For example:
- Suppose A and B are traveling a fixed distance of 120 km.
- A travels at a speed of 60 km/h, so the time taken by A = 120 / 60 = 2 hours.
- B travels at a speed of 40 km/h, so the time taken by B = 120 / 40 = 3 hours.
- The ratio of their speeds is 60:40 or 3:2.
- The ratio of their times is 2:3, showing that time is inversely proportional to speed when distance is constant.
If A starts at 10 PM and moves towards B with a speed of 80 km/h, and B also starts at 10 PM and moves towards A with a speed of 20 km/h, they will meet at point X. At point X, they took the same amount of time. No matter the speed, they will take the same amount of time to meet, but the one with the greater speed will cover more distance. Therefore, A will be closer to B's starting point. This demonstrates that speed impacts distance. In summary, if two bodies start at the same time and move towards each other, the one with the higher speed will cover more distance, and they will meet at the same time.
- Example:
  - A and B start at 10 PM.
  - A's speed = 80 km/h
  - B's speed = 20 km/h
  - Suppose they meet after t hours.
  - The distance covered by A in t hours = 80t km
  - The distance covered by B in t hours = 20t km
  - If the total distance between them is D, then 80t + 20t = D
  - 100t = D
  - So, t = D / 100
  - Both A and B will take the same time t to meet at point X, but A will cover a greater distance (80t km) compared to B (20t km).

Practice Questions:

A and B started driving towards each other from points X and Y at speeds of 72 km/h and 56 km/h, respectively, in order to meet each other. A had traveled 12 km more than B. Find the distance between X and Y.

We know that when they meet, the time taken by both A and B will be the same.
Time = Distance / Speed.
Suppose B traveled 'd' distance, then A will travel (d + 12) distance.
Since time taken by both is the same:
(d + 12) / 72 = d / 56
To solve for 'd', cross-multiply:
(d + 12) * 56 = d * 72
56d + 672 = 72d
72d - 56d = 672
16d = 672
d = 42 km
So, A traveled 42 + 12 = 54 km.
And B traveled 42 km.
Hence, the distance between X and Y is 54 + 42 = 96 km.

A and B start driving from P towards Q. The distance between P and Q is 144 km. The speed of A is 18 km/h less than that of B. After reaching Q, B immediately returns to P and meets A at a distance of 24 km from Q. What is the speed of A?

When they meet, the time taken by both A and B will be the same.
The distance traveled by B when they meet = 144 km (P to Q) + 24 km (returning towards P) = 168 km.
The distance traveled by A when they meet = 144 km (total distance from P to Q) - 24 km (distance from Q) = 120 km.
Let B's speed be x km/h. Then, A's speed will be (x - 18) km/h.
Since the time taken is the same, we can set up the equation:
168 / x = 120 / (x - 18)
Cross-multiplying to solve for x:
168 * (x - 18) = 120 * x
168x - 3024 = 120x
168x - 120x = 3024
48x = 3024
x = 63
Therefore, the speed of A is 63 - 18 = 45 km/h.
Doing it using another method:
We know that if time is constant, then the distance is proportional to speed.
For distances, the ratio of A:B = 120:168.
Upon simplification, we get 5:7.
The difference in the ratio is 2 units.
The difference between A's speed and B's speed is 18 km/h.
So, 2 units = 18 km/h.
1 unit = 9 km/h.
So, A's speed in units is 5.
In km/h, we get 5 * 9 = 45 km/h.
Hence, A's speed is 45 km/h.

A runs 3¹⁄₄ times as fast as B. If A gives B a start of 90m, how far must the winning post be from the starting point of A so that A and B reach it at the same time?

3¹⁄₄ = 13/4
A and B's speed ratio is 13:4.
The difference in speed units is 9.
The difference in distance traveled by A and B is 90m.
So, 9 units = 90m.
1 unit = 10m.
Since time is constant, distance is proportional to speed.
This means A travels 13 units and B travels 4 units.
The distance from A's starting point to the winning post = 13 * 10m = 130m.
Therefore, the winning post must be 130m from A's starting point.

A man can reach a certain place in 60 hours. If he reduces his speed by 1/15th, he goes 20 km less in that time. Find his speed.

Here also, the time is constant.
If he reduces his speed by 1/15th, it means his original speed was 15 units, and his reduced speed is 14 units.
So, the distance will also be proportional. Initially, he traveled 15 units of distance, and with his new reduced speed, he will travel 14 units of distance.
He travels 20 km less, which means 1 unit = 20 km.
So, 15 units = 300 km.
His speed will be 300 km / 60 hours = 5 km/h.

A, while going to school, increases his speed to 13/9th of his actual speed. He reaches his school 24 minutes early. Find A's actual speed, if the distance between his house and school is 13 km.

His actual speed = 9 units and his new speed = 13 units.
The distance is 13 km, which is constant, so speed and time will be inversely proportional.
So, if the speed ratio is 9:13,
The time ratio will be 13:9.
The difference in time units = 4.
4 units = 24 minutes.
1 unit = 6 minutes.
13 units = 78 minutes = 1³⁄₁₀ hours.
So, speed = 13 km / (1³⁄₁₀ hours) = 10 km/h.
Remember: Increase by 13/9 = 9 : 22
and Increase to 13/9 = 9 : 13

A arrives at a place 12 minutes earlier than scheduled time if he rides his bike at 48 km/h. If he rides his bike at 36 km/h, he arrives 18 minutes late. What is the distance of the place from his starting point?

Suppose A's scheduled time was x minutes.
Condition 1: Speed is 48 km/h and time is x - 12 minutes.
Condition 2: Speed is 36 km/h and time is x + 18 minutes.
Speed ratio is 4:3.
As speed and time are inversely proportional (since the distance is the same), the time ratio will be 3:4.
The difference is 1 unit.
Time difference = (x + 18) - (x - 12) = 30 minutes.
So, 1 unit = 30 minutes.
Let's take condition 1:
Speed is 48 km/h and time was 3 units.
3 units = 3 * 30 = 90 minutes = 1.5 hours.
Distance = Speed * Time = 48 km/h * 1.5 hours = 72 km.
Therefore, the distance of the place from A's starting point is 72 km.

In covering a distance of 3600 km, A takes 5 hours more than B. If A increased his speed by 25%, then he would have taken 4 hours less than B. Find the speed of B (in km/h).

A bus travels from X to Y at a constant speed. If its speed were increased by 15 km/h, it would have taken 2 hours less to cover the distance. It would have taken 1 hour more if the speed were decreased by 5 km/h. What is the distance between X and Y?

Let the distance between X and Y be D km.
Let the constant speed be S km/h.
The time to travel the distance at the constant speed is D/S hours.
The time to travel the distance when the speed is increased by 15 km/h is D/(S+15) hours.
Given that when the speed is increased by 15 km/h, it takes 2 hours less:
D/S - D/(S+15) = 2 --- (1)
When the speed is decreased by 5 km/h, it takes 1 hour more:
D/(S-5) - D/S = 1 --- (2)
Now, solve equations (1) and (2) to find D and S:
From equation (1):
(D(S+15) - DS) / (S(S+15)) = 2
(15D) / (S(S+15)) = 2
15D = 2S(S+15) --- (3)
From equation (2):
(D - D(S-5)) / ((S-5)S) = 1
(5D) / ((S-5)S) = 1
5D = S(S-5) --- (4)
From equations (3) and (4):
Set 15D = 2S(S+15) equal to 5D = S(S-5) multiplied by 3 to eliminate D:
15D = 3S^2 - 15S
2S^2 + 30S = 3S^2 - 15S
3S^2 - 2S^2 = 45S
S^2 = 45S
S = 45 km/h (since speed cannot be 0)
Now, substitute S = 45 km/h back into equation (4):
5D = 45^2 - 5 * 45
5D = 2025 - 225
5D = 1800
D = 1800 / 5
D = 360 km
Therefore, the distance between X and Y is 360 km.
We also have a short formula to find the distance:
D = ((constant speed * modified speed) / difference between two speeds) * difference in time
When the speed is increased by 15 km/h:
D = ((S * (S + 15)) / 15) * 2
When the speed is decreased by 5 km/h:
D = ((S * (S - 5)) / 5) * 1
We can equate them both now:
((S * (S + 15)) / 15) * 2 = ((S * (S - 5)) / 5) * 1
Cancel out the common terms:
(S * (S + 15)) * 2 / 15 = (S * (S - 5)) / 5
(S * (S + 15)) * 2 = (S * (S - 5)) * 3
S * (S + 15) * 2 = S * (S - 5) * 3
(S + 15) * 2 = (S - 5) * 3
2S + 30 = 3S - 15
Solving the equation:
2S + 30 = 3S - 15
30 + 15 = 3S - 2S
45 = S
So, S = 45 km/h.
Now, using the speed S to find the distance D:
D = ((45 * (45 + 15)) / 15) * 2
D = (45 * 60 / 15) * 2
D = (2700 / 15) * 2
D = 180 * 2
D = 360 km
Therefore, the distance between X and Y is 360 km.

Relative Speed

When two bodies are in motion simultaneously, we don't consider their individual speeds. Instead, we consider their relative speed. Relative speed can be of two types:
1. When two bodies are traveling in opposite directions, the relative speed is the sum of their respective speeds: s1 + s2.
2. When two bodies are traveling in the same direction, the relative speed is the difference of their respective speeds: s1 - s2.
Distance:
- If the point object is a man, pole, or tree, then the distance is the train's own length as the length of the man, pole, or tree is negligible.
- If it is a platform, tunnel, or bridge, then the distance is the train's own length plus the length of the platform, tunnel, or bridge.
- If it is another train, then the distance is the sum of the lengths of both trains.
We use this when calculating the time taken for a train to pass a stationary object or another moving object.
Conversion:
- To convert from m/s to km/h, multiply by 18/5.
- To convert from km/h to m/s, multiply by 5/18.

How many seconds will a train 100m long running at the rate of 36 km/h take to pass a stationary pole?

When a train passes a point object, it covers its own length.
Distance = 100 meters.
Speed = 36 km/h.
Convert speed from km/h to m/s:
Speed = 36 * (5/18) = 10 m/s.
Time = Distance / Speed:
Time = 100 meters / 10 m/s = 10 seconds.
Therefore, the train will take 10 seconds to pass the stationary pole.

How many seconds will a train 110m long running at the rate of 72 km/h take to cross a bridge 130m long?

Distance = length of train + length of bridge:
Distance = 110 meters + 130 meters = 240 meters.
Convert speed from km/h to m/s:
Speed = 72 * (5/18) = 20 m/s.
Time = Distance / Speed:
Time = 240 meters / 20 m/s = 12 seconds.
Therefore, the train will take 12 seconds to cross the bridge.

Two trains 121m and 99m in length respectively are running in opposite directions, one at the rate of 40 km/h and the other at the rate of 32 km/h. In what time will they completely cross each other from the moment they meet?

Total distance = length of first train + length of second train:
Total distance = 121 meters + 99 meters = 220 meters.
Since they are moving in opposite directions, their relative speed is the sum of their speeds:
Relative speed = 40 km/h + 32 km/h = 72 km/h.
Convert the relative speed from km/h to m/s:
Relative speed = 72 * (5/18) = 20 m/s.
Time = Total distance / Relative speed:
Time = 220 meters / 20 m/s = 11 seconds.
Therefore, the two trains will completely cross each other in 11 seconds.

Two trains 110m and 90m in length respectively are running in the same direction, one at the rate of 50 km/h and the other at the rate of 68 km/h. In what time will they completely cross each other?

Total distance = length of first train + length of second train:
Total distance = 110 meters + 90 meters = 200 meters.
Since they are moving in the same direction, their relative speed is the difference of their speeds:
Relative speed = 68 km/h - 50 km/h = 18 km/h.
Convert the relative speed from km/h to m/s:
Relative speed = 18 * (5/18) = 5 m/s.
Time = Total distance / Relative speed:
Time = 200 meters / 5 m/s = 40 seconds.
Therefore, the two trains will completely cross each other in 40 seconds.

Two trains are moving in the same direction at 50 km/h and 30 km/h. The faster train crosses a man in the slower train in 18 seconds. Find the length of the faster train.

Total distance = Length of the faster train = D
Relative speed = Speed of the faster train - Speed of the slower train:
Relative speed = 50 km/h - 30 km/h = 20 km/h.
Convert the relative speed from km/h to m/s:
Relative speed = 20 * (5/18) = 5.56 m/s.
Time = 18 seconds.
Distance = Speed * Time:
D = 5.56 m/s * 18 s = 100 meters.
Therefore, the length of the faster train is 100 meters.

A 150m long train crosses a platform in 45 seconds and crosses a man in 9 seconds. Find the length of the platform.

Total distance = Train length + Platform length = 150 meters + P meters.
Time to cross the platform = 45 seconds.
Using the man example:
Distance = 150 meters (train length) and time = 9 seconds.
Speed of the train = 150 meters / 9 seconds = 16.67 m/s.
Now, using the speed to find the platform length:
Total distance to cross the platform = Speed * Time:
Total distance = 16.67 m/s * 45 seconds = 750 meters.
So, 150 meters (train length) + P meters (platform length) = 750 meters.
Platform length, P = 750 meters - 150 meters = 600 meters.
Therefore, the length of the platform is 600 meters.

Time, Speed & Distance

Time, Speed & Distance

Relation Between Time, Speed, and Distance

Practice Questions:

A and B started driving towards each other from points X and Y at speeds of 72 km/h and 56 km/h, respectively, in order to meet each other. A had traveled 12 km more than B. Find the distance between X and Y.

A and B start driving from P towards Q. The distance between P and Q is 144 km. The speed of A is 18 km/h less than that of B. After reaching Q, B immediately returns to P and meets A at a distance of 24 km from Q. What is the speed of A?

A runs 31⁄4 times as fast as B. If A gives B a start of 90m, how far must the winning post be from the starting point of A so that A and B reach it at the same time?

A man can reach a certain place in 60 hours. If he reduces his speed by 1/15th, he goes 20 km less in that time. Find his speed.

A, while going to school, increases his speed to 13/9th of his actual speed. He reaches his school 24 minutes early. Find A's actual speed, if the distance between his house and school is 13 km.

A arrives at a place 12 minutes earlier than scheduled time if he rides his bike at 48 km/h. If he rides his bike at 36 km/h, he arrives 18 minutes late. What is the distance of the place from his starting point?

In covering a distance of 3600 km, A takes 5 hours more than B. If A increased his speed by 25%, then he would have taken 4 hours less than B. Find the speed of B (in km/h).

A bus travels from X to Y at a constant speed. If its speed were increased by 15 km/h, it would have taken 2 hours less to cover the distance. It would have taken 1 hour more if the speed were decreased by 5 km/h. What is the distance between X and Y?

Relative Speed

How many seconds will a train 100m long running at the rate of 36 km/h take to pass a stationary pole?

How many seconds will a train 110m long running at the rate of 72 km/h take to cross a bridge 130m long?

Two trains 121m and 99m in length respectively are running in opposite directions, one at the rate of 40 km/h and the other at the rate of 32 km/h. In what time will they completely cross each other from the moment they meet?

Two trains 110m and 90m in length respectively are running in the same direction, one at the rate of 50 km/h and the other at the rate of 68 km/h. In what time will they completely cross each other?

Two trains are moving in the same direction at 50 km/h and 30 km/h. The faster train crosses a man in the slower train in 18 seconds. Find the length of the faster train.

A 150m long train crosses a platform in 45 seconds and crosses a man in 9 seconds. Find the length of the platform.

A runs 3¹⁄₄ times as fast as B. If A gives B a start of 90m, how far must the winning post be from the starting point of A so that A and B reach it at the same time?