Unit 4: Time, Distance, Trains, Boats and Streams - Subjective Questions

Definitions:

• Distance: The total length of the path travelled by an object. It is commonly measured in metres or kilometres.
• Time: The duration taken by an object to cover a distance. It is commonly measured in seconds, minutes, or hours.
• Speed: The distance travelled by an object per unit of time.

Fundamental relationship:

From this formula, the other relationships are:

For example, if a vehicle covers km in hours, its speed is:

Since km m and hour seconds:

Therefore:

• To convert km/h into m/s, multiply by .
• To convert m/s into km/h, multiply by .

Conversion of km/h:

Conversion of m/s:

Thus, km/h equals m/s, while m/s equals km/h.

The relationship among speed, distance, and time is:

The relevant proportionalities are:

• For constant time, distance is directly proportional to speed: .
• For constant speed, distance is directly proportional to time: .
• For constant distance, time is inversely proportional to speed: .

If the distance remains constant and the speed changes from to , then:

Therefore:

For example, if speed is doubled, the time required becomes half. If speed is increased three times, the time becomes one-third of the original time.

First part of the journey:

Second part of the journey:

Therefore:

• Total distance km
• Total time hours

Average speed is calculated using total distance and total time:

Thus, the student takes hours, and the average speed for the entire journey is km/h.

Let each of the two equal distances be km.

Total distance:

Time taken for the first distance:

Time taken for the second distance:

Therefore, total time is:

Average speed is:

After simplification:

This is the harmonic mean of the two speeds. It applies only when the distances travelled at the two speeds are equal.

Average speed is defined as:

The arithmetic mean of two speeds and is:

These two expressions are not always equal because average speed depends on both the distance covered and the time spent at each speed.

• If an object travels for equal intervals of time at speeds and , its average speed is .
• If it travels equal distances at the two speeds, its average speed is .

Therefore, average speed equals the arithmetic mean when the object travels at the given speeds for equal durations. The two formulas also give the same value when .

Let the fixed distance be km.

Time taken at km/h:

Time taken at km/h:

The difference is minutes:

Therefore:

Taking the least common multiple:

Hence, the distance covered is km.

Relative speed is the speed of one moving object as observed from another moving object.

Suppose two objects move with speeds and , where .

Same direction:

Their relative speed is the difference of their speeds:

This is because the faster object gains only units of distance per unit time on the slower object.

Opposite directions:

Their relative speed is the sum of their speeds:

This is because both objects reduce the distance between them simultaneously.

For example, for speeds km/h and km/h:

• Same direction relative speed km/h.
• Opposite direction relative speed km/h.

Since the cyclists move towards each other, their relative speed is:

Time required to meet:

Distance covered by the first cyclist:

Distance covered by the second cyclist:

Verification:

Thus, they meet after hours, at a point km from the starting position of the first cyclist and km from the starting position of the second cyclist.

When a train crosses a stationary person, pole, or signal, it must cover a distance equal to its own length.

The required formula is:

First, convert km/h into m/s:

The train length is m. Therefore:

Hence, the train takes seconds to cross the pole.

To completely cross a platform, the front of the train must travel from the beginning of the platform until the rear of the train passes the far end. Hence, the total distance covered is the sum of the train and platform lengths.

Convert the train's speed into m/s:

Now calculate the time:

Therefore, the train takes seconds to cross the platform completely.

When two trains cross each other, the total distance to be covered is the sum of their lengths:

Since they move in opposite directions, their relative speed is the sum of their speeds:

Convert relative speed into m/s:

Therefore, the crossing time is:

Thus, the two trains cross each other completely in seconds.

For trains moving in the same direction, relative speed is the difference of their speeds:

Convert it into m/s:

To overtake completely, the faster train must cover a relative distance equal to the sum of both train lengths:

Therefore:

Hence, the faster train takes seconds to overtake the slower train completely.

Let the train's length be metres and speed be m/s.

When the train crosses a pole:

Therefore:

When it crosses the m platform:

Substitute :

Convert the speed into km/h:

The train's length is:

Thus, the train is m long and travels at km/h.

Let the speed of a boat in still water be and the speed of the stream be .

• Downstream movement: The boat moves in the direction of the stream. The stream assists the boat.
• Upstream movement: The boat moves against the direction of the stream. The stream opposes the boat.
• Speed in still water: The boat's speed when there is no current.
• Speed of stream: The speed at which the water current flows.

The formulas are:

If downstream speed is and upstream speed is , then:

These relationships assume that the boat and stream speeds remain constant.

Let downstream speed be km/h and upstream speed be km/h.

The speed of the boat in still water is:

The speed of the stream is:

Verification:

• Downstream speed km/h.
• Upstream speed km/h.

Thus, the boat's speed in still water is km/h, and the stream's speed is km/h.

The downstream speed is:

Time taken to travel km downstream:

The upstream speed is:

Time taken to travel km upstream:

Therefore, total time is:

Hence, the complete downstream and upstream journey takes hours.

Let the one-way distance be .

The downstream and upstream speeds are:

Time taken downstream:

Time taken upstream:

Total distance is , and total time is:

Therefore, average speed is:

This value is less than whenever , because the extra upstream time is greater than the time saved downstream.

Downstream speed is:

Upstream speed is:

Speed of the boat in still water:

Speed of the stream:

For the round trip:

• Total distance km
• Total time hours

Therefore:

Thus, the boat's speed in still water is km/h, the stream's speed is km/h, and the round-trip average speed is approximately km/h.

Let the distance be km and the scheduled travel time be hours.

At km/h, the man is minutes late:

At km/h, he is minutes early:

Subtracting the second equation from the first:

Therefore:

Using the second condition:

Since hour equals minutes, the scheduled travel time is hours minutes.

Thus, the destination is km away, and the scheduled travel time is hours minutes.