Unit 4: Time, Speed and Distance - Subjective Questions

Definitions:

• Distance: The total length of the path travelled by an object.
• Time: The duration required to complete a journey.
• Speed: The distance travelled per unit of time.

Relationship:

The formula can be rearranged as:

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

The basic unit relationships are:

• km m
• hour seconds

Therefore:

Hence:

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

Examples:

At constant time: Distance is directly proportional to speed.

Thus, if speed doubles while time remains fixed, the distance also doubles.

At constant speed: Distance is directly proportional to time.

Thus, travelling for three times as long produces three times the distance.

At constant distance: Speed is inversely proportional to time.

Therefore:

If speed increases, the time required to cover the same distance decreases in the same ratio.

Average speed is the ratio of the total distance travelled to the total time taken:

It is not always equal to the arithmetic mean of the speeds. The arithmetic mean:

is valid when an object travels at the two speeds for equal intervals of time.

When equal distances are covered at different speeds, the times are unequal. In that case, the average speed is:

Thus, average speed must normally be calculated using total distance and total time.

Let each part of the journey have distance .

Step 1: Calculate the total distance.

Step 2: Calculate the time taken for each part.

Step 3: Calculate total time.

Step 4: Apply the average-speed formula.

Therefore:

This is the harmonic mean of the two speeds.

First part:

Second part:

Total distance:

Total time:

Therefore, the average speed is:

Since the time intervals are equal, this also equals the arithmetic mean of km/h and km/h.

For a fixed distance, speed and time are inversely proportional:

Substituting the values:

Therefore:

Alternatively, the distance is:

At km/h, the required time is:

The higher speed reduces the time needed to cover the same distance.

A pole or standing person has negligible length. Therefore, the train must travel a distance equal to its own length to cross it completely.

If the length of the train is metres and its speed is m/s, then:

Example: A m train moving at km/h has speed:

The time required to cross a pole is:

The speed should be converted into m/s when length is measured in metres.

To cross the platform completely, the train must cover the sum of its own length and the platform's length.

Total distance:

Convert the speed:

Calculate the time:

Thus, the train takes:

When two trains move in the same direction, their relative speed is the difference of their speeds.

Relative speed:

To overtake completely, the relative distance covered is the sum of the train lengths:

Therefore:

The trains' individual ground speeds are replaced by their relative speed for this calculation.

For objects moving in opposite directions, the relative speed is the sum of their speeds.

Relative speed:

Relative distance:

Time required:

Both train lengths are included because they must pass each other completely.

Relative speed is the speed at which the distance between two moving objects changes, as observed from one of the objects.

If two objects have speeds and :

• Same direction:
  
  The faster object gains on the slower object at the difference of their speeds.

• Opposite directions:
  
  The distance between them changes at the sum of their speeds.

Relative speed is used in:

• Train-crossing and overtaking problems
• Meeting-point problems
• Pursuit problems
• Motion on parallel tracks or roads

The train covers its own length relative to the walking man.

Relative speed:

Convert this speed into km/h:

Since the train and the man move in the same direction:

Therefore:

The man's speed is added because the calculated relative speed is the difference between the two actual speeds.

• Speed in still water: The speed of a boat when there is no water current. Let it be .
• Speed of stream: The speed at which the current flows. Let it be .
• Downstream speed: The boat moves in the direction of the current, so the current increases its effective speed:
  
• Upstream speed: The boat moves against the current, so the current decreases its effective speed:
  

Usually, must hold for the boat to travel upstream. Downstream travel is faster, while upstream travel takes more time for the same distance.

Let:

• be the speed of the boat in still water.
• be the speed of the stream.
• be the downstream speed.
• be the upstream speed.

The basic equations are:

Adding the equations:

Therefore:

Subtracting the equations:

Therefore:

Thus, the boat's speed is half the sum of upstream and downstream speeds, while the stream's speed is half their difference.

Given:

Speed of the boat in still water:

Speed of the stream:

Therefore:

• Boat's speed in still water
• Stream's speed

Downstream speed:

Upstream speed:

Downstream time:

Upstream time:

Total time:

Total distance:

Average speed:

Thus, the total travel time is .

Since the cars move toward each other, their relative speed is:

Meeting time:

Distance travelled by the first car:

Distance travelled by the second car:

Verification:

Therefore, the cars meet after , at a point from the city where the first car started.

Total moving time:

Distance covered while moving:

Total stoppage time:

Total elapsed time:

Average speed:

Stoppage time is included in total time even though no distance is covered during the stops.

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

When the train crosses a pole:

Therefore:

When it crosses the bridge, the total distance is :

Substitute :

Convert the speed into km/h:

Train length:

Therefore:

• Length of train:
• Speed of train: