Time, Speed, and Distance

Time, Speed, and Distance: Basic Concepts and Formulae

The relationship between Time, Speed, and Distance is a fundamental concept used to describe the motion of objects. These three quantities are interconnected, and understanding their relationship is crucial for solving problems involving motion.

Definitions:

• Distance: Distance is the total length of the path covered by a moving object during its journey. It is a scalar quantity, meaning it only has magnitude, not direction. Common units of distance include metres (m), kilometres (km), miles, feet, etc.
• Time: Time refers to the duration for which the motion takes place. It is also a scalar quantity. Common units of time include seconds (s), minutes, hours (h), days, etc.
• Speed: Speed is the rate at which an object moves or covers distance. It is defined as the distance covered by an object per unit of time. Speed is also a scalar quantity.

Relationship and Formulae:

The relationship between Distance, Speed, and Time can be expressed by a single fundamental formula, from which the other two can be derived:

The defining formula for Speed is:

From this basic formula, we can rearrange the terms to find formulae for Distance and Time:

To find Distance, multiply Speed by Time:

To find Time, divide Distance by Speed:

These three formulae are the bedrock of solving problems in Time, Speed, and Distance. It is extremely important to ensure that the units of Distance, Speed, and Time are consistent when using these formulae. For example, if Speed is in km/h, Distance should be in km and Time in hours. If Speed is in m/s, Distance should be in m and Time in seconds.

Constant Speed (Uniform Motion):

The formulas presented above are most directly applicable when the speed of the object is constant throughout the motion. This is known as uniform motion. In many problems, we assume constant speed unless stated otherwise. If the speed changes during different parts of the journey, we might need to use these formulas for each segment of the journey where the speed is constant, or we introduce the concept of Average Speed, which will be discussed later.

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Unit Conversions for Time, Speed, and Distance

In Time, Speed, and Distance problems, quantities are often given in different units. To perform calculations using the standard formulae ($\text{D} = \text{S} \times \text{T}$, etc.), it is essential to convert all quantities to consistent units. The most common units encountered are kilometres per hour (km/h) for speed, metres per second (m/s) for speed, kilometres (km) or metres (m) for distance, and hours (h) or seconds (s) for time.

Key Unit Conversions:

Distance Units:

• 1 kilometre (km) $= 1000$ metres (m)

Time Units:

• 1 hour (h) $= 60$ minutes
• 1 minute $= 60$ seconds (s)
• 1 hour (h) $= 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = 3600$ seconds (s)

Speed Units: Converting between km/h and m/s:

This is a very frequent conversion in problems.

To convert speed from kilometres per hour (km/h) to metres per second (m/s):

Consider a speed of 1 km/h. We convert the distance (km) to metres (m) and the time (h) to seconds (s).

Convert units:

Simplify the fraction $\frac{1000}{3600}$:

So, to convert any speed $x$ from km/h to m/s, multiply by $\frac{5}{18}$:

To convert speed from metres per second (m/s) to kilometres per hour (km/h):

We can rearrange the previous relationship: $\frac{5}{18} \text{ m/s} = 1 \text{ km/h}$.

To find the conversion factor for 1 m/s, divide 1 by $\frac{5}{18}$:

So, to convert any speed $x$ from m/s to km/h, multiply by $\frac{18}{5}$:

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Problems involving Average Speed

When an object travels at different speeds over different segments of a journey, or if its speed is not constant throughout, the concept of Average Speed is used to describe the overall rate of motion for the entire trip. Average speed is defined as the total distance covered divided by the total time taken for the entire journey.

Formula for Average Speed:

The fundamental formula for calculating average speed is:

It is important to note that average speed is generally not the simple arithmetic mean of the different speeds involved, unless specific conditions are met (as shown in the special cases below).

Special Cases for Calculating Average Speed:

There are two common scenarios where simplified formulas for average speed can be derived:

Case 1: When the time taken for different parts of the journey is the same.

If an object travels at speed $S_1$ for a time duration $T$ and then travels at speed $S_2$ for the same time duration $T$.

Distance covered in the first part ($D_1$) $= S_1 \times T$.

Distance covered in the second part ($D_2$) $= S_2 \times T$.

Total Distance $= D_1 + D_2 = (S_1 \times T) + (S_2 \times T) = T(S_1 + S_2)$.

Total Time Taken $= T + T = 2T$.

Using the general average speed formula (vi):

Assuming $T \neq 0$, we can cancel T:

In this specific case where the time taken for each part of the journey is equal, the average speed is the simple arithmetic mean of the speeds. This formula can be extended: if an object travels at speeds $S_1, S_2, \dots, S_n$ for the same time duration in each segment, the average speed is $\frac{S_1 + S_2 + \dots + S_n}{n}$.

Case 2: When the distance covered for different parts of the journey is the same.

If an object covers a distance $D$ at a speed $S_1$ and then covers the same distance $D$ at a speed $S_2$.

Time taken for the first part ($T_1$) $= \frac{\text{Distance}}{\text{Speed}_1} = \frac{D}{S_1}$.

Time taken for the second part ($T_2$) $= \frac{\text{Distance}}{\text{Speed}_2} = \frac{D}{S_2}$.

Total Distance $= D + D = 2D$.

Total Time Taken $= T_1 + T_2 = \frac{D}{S_1} + \frac{D}{S_2}$.

Combine the terms in Total Time using a common denominator:

Using the general average speed formula (vi):

Assuming $D \neq 0$, we can cancel D and rearrange the fraction:

This formula, which is the harmonic mean of the two speeds, is used when an object covers the same distance at two different speeds. It can be extended for more than two speeds if the distance covered at each speed is the same. For three equal distances covered at speeds $S_1, S_2, S_3$, the average speed is $\frac{3}{\left(\frac{1}{S_1} + \frac{1}{S_2} + \frac{1}{S_3}\right)} = \frac{3 S_1 S_2 S_3}{S_1 S_2 + S_2 S_3 + S_3 S_1}$.

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Problems involving Relative Speed

When two or more objects are in motion, their relative speed is the speed of one object as observed from the frame of reference of another moving object. The concept of relative speed simplifies problems where we need to determine the time taken for objects to meet, overtake each other, or cover a certain distance relative to each other.

Calculating Relative Speed:

The calculation of relative speed depends on the direction of motion of the objects.

Let the speeds of two objects be $S_1$ and $S_2$.

• Case 1: Objects are moving in the same direction.

  When two objects move in the same direction, their relative speed is the absolute difference between their speeds. This represents the rate at which the distance between them increases (if the leading object is faster) or decreases (if the trailing object is faster).

  $\boldsymbol{\text{Relative Speed} = |S_1 - S_2|}$

  ... (ix)

  Typically, this is calculated as (Speed of Faster Object) - (Speed of Slower Object) when one object is trying to catch up to another.

• Case 2: Objects are moving in opposite directions.

  When two objects move towards each other or away from each other, their relative speed is the sum of their individual speeds. This represents the rate at which the distance between them decreases (when moving towards each other) or increases (when moving away from each other).

  $\boldsymbol{\text{Relative Speed} = S_1 + S_2}$

  ... (x)

Note: Ensure that $S_1$ and $S_2$ are in consistent units before calculating relative speed.

Using Relative Speed:

Once the relative speed is calculated, problems can often be solved by using the basic Time = Distance / Speed formula, where the 'Speed' is the Relative Speed and the 'Distance' is the initial distance between the objects that needs to be covered (or the distance gained/lost).

Common scenarios include:

• Time to meet = $\frac{\text{Initial distance between them}}{\text{Relative speed (opposite directions)}}$
• Time to overtake = $\frac{\text{Initial distance between them}}{\text{Relative speed (same direction)}}$

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Problems on Trains

Problems involving trains are a common category within Time, Speed, and Distance. These problems often introduce the concept of the length of the moving object (the train) itself, which needs to be considered when the train crosses another object or covers a certain distance.

Train Crossing a Point Object:

A point object is an object whose length or size is negligible compared to the length of the train. Examples include a stationary pole, a standing person, a signal post, or a milestone. When a train crosses a point object, the distance the train needs to cover is equal to its own length.

Imagine the front of the train reaching the pole and the train finishing crossing when the rear of the train leaves the pole. The distance covered by any point on the train (e.g., the front tip) during this process is exactly the length of the train.

Using the basic formula Time = Distance / Speed:

Train Crossing an Object with Length:

An object with length could be a platform, a bridge, a tunnel, or another train (stationary or moving). When a train crosses an object that has a significant length, the distance the train needs to cover is the sum of its own length and the length of the object it is crossing.

Imagine the front of the train entering the object (e.g., platform). To completely cross the object, the rear of the train must leave the object. By the time the rear of the train leaves the object, the front of the train will have covered a distance equal to the length of the object plus the length of the train itself.

Using the basic formula Time = Distance / Speed:

Train Crossing a Moving Object:

When a train crosses a moving object, such as another train or a person running alongside or towards the train, the concept of relative speed becomes essential. The distance covered for the crossing depends on whether the moving object also has a significant length.

• Case A: Train crosses another moving train.

  The distance to be covered for the crossing is the sum of the lengths of the two trains.

  Distance $= \text{Length of Train 1} + \text{Length of Train 2}$

  The speed at which this distance is covered is the relative speed of the two trains.

  • If trains are moving in the same direction: Relative Speed $= |S_1 - S_2|$
  • If trains are moving in opposite directions: Relative Speed $= S_1 + S_2$

  Using Time = Distance / Speed:

  $\boldsymbol{\text{Time} = \frac{\text{Sum of Lengths of Two Trains}}{\text{Relative Speed}}}$

  ... (xiii)

• Case B: Train crosses a moving person.

  A moving person is considered a point object (negligible length compared to the train). The distance to be covered for the crossing is the length of the train itself.

  Distance $= \text{Length of the Train}$

  The speed at which this distance is covered is the relative speed of the train and the person.

  • If the train and person are moving in the same direction: Relative Speed $= \text{Speed of Train} - \text{Speed of Person}$ (assuming train is faster).
  • If the train and person are moving in opposite directions: Relative Speed $= \text{Speed of Train} + \text{Speed of Person}$.

  Using Time = Distance / Speed:

  $\boldsymbol{\text{Time} = \frac{\text{Length of the Train}}{\text{Relative Speed}}}$

  ... (xiv)

Important: Before applying any formula, ensure all quantities (lengths, speeds) are in consistent units (e.g., convert km/h to m/s, km to m, hours to seconds if needed).

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Problems on Boats and Streams (Upstream and Downstream)

Problems involving boats and streams (or swimmers in a river) are a specialized category of Time, Speed, and Distance problems. They involve understanding how the motion of the water (the stream or current) affects the speed of the boat or person moving in it.

Key Terms:

• Speed of the boat or man in still water ($S_b$ or $V_b$): This is the speed at which the boat or a person can travel in water that is not flowing. It represents the boat's or person's own effort or capacity for speed, unaffected by the current.
• Speed of the stream or current ($S_s$ or $V_s$): This is the speed at which the water is flowing.

Speeds in Different Directions Relative to the Stream:

The speed of the stream either helps or opposes the motion of the boat/man, depending on the direction of travel.

• Speed Downstream ($S_d$): When the boat or person moves in the same direction as the stream, the stream's speed adds to their speed in still water. The effective speed becomes the sum of the speed in still water and the speed of the stream.

  $\boldsymbol{\text{Speed Downstream} = \text{Speed in Still Water} + \text{Speed of Stream}}$

  $\boldsymbol{S_d = S_b + S_s}$

  ... (i)

• Speed Upstream ($S_u$): When the boat or person moves in the opposite direction to the stream, the stream's speed acts against their motion, reducing their effective speed. The effective speed becomes the difference between the speed in still water and the speed of the stream.

  $\boldsymbol{\text{Speed Upstream} = \text{Speed in Still Water} - \text{Speed of Stream}}$

  $\boldsymbol{S_u = S_b - S_s}$

  ... (ii)

  Note: For motion upstream to be possible, the speed of the boat/man in still water ($S_b$) must be greater than the speed of the stream ($S_s$). If $S_b \le S_s$, the boat/man would be carried downstream by the current.

Finding Speed of Boat ($S_b$) and Speed of Stream ($S_s$) from Downstream and Upstream Speeds:

Often, problems provide the speed of the boat/man downstream and upstream, and you need to find their speeds in still water or the speed of the stream. We can derive formulae for this by solving the two equations above simultaneously.

We have the system of equations:

Adding Equation 1 and Equation 2:

Solving for $S_b$:

Subtracting Equation 2 from Equation 1:

Solving for $S_s$:

These two formulas (iii and iv) are very useful for quickly finding the speed in still water and the speed of the stream when the downstream and upstream speeds are known.

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Problems on Races and Games

Problems categorized under Races and Games are specific applications of Time, Speed, and Distance concepts, focusing on competitive events. These problems often involve comparing the speeds of participants, understanding the effect of 'starts' given by one participant to another, and determining race outcomes.

Key Concepts:

• Race Distance: The total length of the track or path over which the race is conducted.
• Start: An advantage given by a faster participant to a slower one to make the competition fairer or more interesting.
  • Start by Distance: If A gives B a start of $x$ metres in a race of length $L$ metres, it means A starts from the beginning line, while B starts $x$ metres ahead of the beginning line. For the race to end at the same time (a dead heat with the start), A must cover the full distance $L$, while B only needs to cover $(L - x)$ metres. If they started simultaneously, they take the same amount of time.
  • Start by Time: If A gives B a start of $t$ seconds, it means B starts the race $t$ seconds before A starts. If they finish simultaneously, it means B runs for the total duration of the race ($T$), while A runs for $(T - t)$ seconds.
• Beating in a Race (by Distance): If participant A beats participant B by $x$ metres in a race of length $L$ metres, it means when A finishes the race (covers $L$ metres), B is still $x$ metres away from the finishing line (B has covered $L - x$ metres). If they start simultaneously, they run for the same amount of time until the winner finishes. In this case, since Time is constant, the ratio of distances covered is equal to the ratio of their speeds. $\frac{\text{Speed of A}}{\text{Speed of B}} = \frac{L}{L-x}$.
• Beating in a Race (by Time): If participant A beats participant B by $t$ seconds in a race of length $L$ metres, it means A completes the race $t$ seconds faster than B. If they started simultaneously, Time taken by A ($T_A$) $= \frac{L}{\text{Speed of A}}$ and Time taken by B ($T_B$) $= \frac{L}{\text{Speed of B}}$. Then, $T_B - T_A = t$. When Distance is constant, Speed is inversely proportional to Time. $\frac{\text{Speed of A}}{\text{Speed of B}} = \frac{T_B}{T_A}$.
• Dead Heat: A dead heat occurs when all participants finish the race at the exact same moment. In problems involving starts, a dead heat implies that the advantage given by the start is perfectly offset by the speed difference.

These problems often rely on the relationships derived from the basic Time, Speed, Distance formulas, particularly when one of the quantities (Time or Distance) is constant:

• If Time is constant, Speed $\propto$ Distance. Ratio of Speeds = Ratio of Distances Covered in that time.
• If Distance is constant, Speed $\propto \frac{1}{\text{Time}}$. Ratio of Speeds = Inverse Ratio of Times taken to cover that distance.

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Advanced Time, Speed, and Distance Problems

Advanced problems in Time, Speed, and Distance often combine multiple concepts, involve scenarios with varying start times or places, or require applying TSD principles to situations like circular tracks. These problems demand a thorough understanding of the basic formulas, relative speed, and careful problem-solving strategies.

Meeting Points - Different Start Times:

If two individuals or vehicles start moving towards each other from two different points at different times, you cannot directly use the relative speed with the total distance. The typical approach is to first calculate the distance covered by the entity that started earlier, until the time the second entity starts. Once both entities are moving simultaneously, calculate the remaining distance between them. This remaining distance is then covered by both entities moving towards each other at their combined relative speed (sum of speeds).

The actual meeting time is the second entity's start time plus the calculated time taken to meet.

Circular Tracks:

Problems involving motion on a circular track introduce scenarios where objects moving in the same or opposite directions may meet at various points. The distance considered for meeting points is related to the length of the track.

Let the length of the circular track be $L$, and the speeds of two people be $S_1$ and $S_2$.

• Meeting for the First Time (Starting from the same point):
  • Moving in Opposite Directions: They meet for the first time when the sum of the distances they have covered equals the length of the track ($L$). The relative speed is $S_1 + S_2$.

    $\boldsymbol{\text{Time to meet for 1st time} = \frac{\text{Length of Track}}{\text{Relative Speed (Sum)}} = \frac{L}{S_1 + S_2}}$

    ... (xvi)

  • Moving in the Same Direction: The faster person gains on the slower person. They meet for the first time when the faster person has gained a distance equal to the length of the track ($L$) over the slower person. The relative speed is $|S_1 - S_2|$.

    $\boldsymbol{\text{Time to meet for 1st time} = \frac{\text{Length of Track}}{\text{Relative Speed (Difference)}} = \frac{L}{|S_1 - S_2|}}$

    ... (xvii)

• Meeting at the Starting Point for the First Time: This occurs after a time that is the LCM of the individual times taken by each person to complete one full lap.

  Time for A to complete 1 lap $= \frac{L}{S_1}$

  Time for B to complete 1 lap $= \frac{L}{S_2}$

  $\boldsymbol{\text{Time to meet at starting point for 1st time} = \text{LCM}\left(\frac{L}{S_1}, \frac{L}{S_2}\right)}$

  ... (xviii)

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