Train Speeds Calculation: Opposite and Same Direction Crossings
Let the lengths of the two trains be $L_1$ and $L_2$ meters, and their respective speeds be $S_1$ and $S_2$ meters per second (m/s).
When two trains cross each other, the total distance involved is the sum of their lengths ($L_1 + L_2$). The relationship is given by: Distance = Speed $\times$ Time.
Scenario 1: Crossing in Opposite Directions
When the trains move towards each other, their relative speed is the sum of their speeds: $S_{relative} = S_1 + S_2$. The time taken to cross is 15 seconds.
Using the distance formula:
$ L_1 + L_2 = (S_1 + S_2) \times 15 \quad \text{(Equation 1)} $Scenario 2: Crossing in Same Direction
When the trains move in the same direction, their relative speed is the difference between their speeds: $S_{relative} = |S_1 - S_2|$. Assume $S_1 > S_2$, so the relative speed is $S_1 - S_2$. The time taken to cross is 45 seconds.
Using the distance formula:
$ L_1 + L_2 = (S_1 - S_2) \times 45 \quad \text{(Equation 2)} $Deriving the Speed Relationship
Equate the expressions for the total length ($L_1 + L_2$) from Equation 1 and Equation 2:
$ 15(S_1 + S_2) = 45(S_1 - S_2) $Divide both sides by 15:
$ S_1 + S_2 = 3(S_1 - S_2) $Simplify the equation:
$ S_1 + S_2 = 3S_1 - 3S_2 $Rearrange the terms to find the relationship between $S_1$ and $S_2$:
$ S_2 + 3S_2 = 3S_1 - S_1 $ $ 4S_2 = 2S_1 $ $ S_1 = 2S_2 $This implies that the speed of one train is twice the speed of the other.
Checking the Options
We need to find the pair of speeds from the options where one speed is double the other:
- Option 1: 20 m/s and 45 m/s ($45 \neq 2 \times 20$)
- Option 2: 15 m/s and 45 m/s ($45 \neq 2 \times 15$)
- Option 3: 30 m/s and 60 m/s ($60 = 2 \times 30$). This satisfies the condition $S_1 = 2S_2$.
- Option 4: 30 m/s and 50 m/s ($50 \neq 2 \times 30$)
Conclusion
The pair of speeds that satisfies the condition derived from the crossing times is 30 m/s and 60 m/s.
Important Questions from Speed Time & Distance (Notes)
A and B have to travel from place P to place Q following the same route in their respective cars. A drives at $60$ kmph while B drives at $80$ kmph. Find the time taken by B to reach place Q if A takes $12$ hrs.
On a straight road, a bus is $60$ km ahead of a car running in the same direction. After $3$ hours, the car is $90$ km ahead of the bus. If the speed of the bus is $45$ km/h, then what is the speed of the car (in km/h)?
A train running at the speed of $90$ kmph crosses a $250$ m long platform in $26$ seconds. What is the length of the train (in m)?
A car covers 4 successive stretches of 3 km each at speed of 10 kmph, 20 kmph, 30 kmph and 60 kmph respectively. The average speed of the car for the entire journey is:
- A car travels a total distance L. It travels half the distance with speed $v_1$ and the other half with speed $v_2$. The average speed of the car is :
