Question 1

A train covers a distance of 12 km in 10 minutes. If it takes 6 seconds to pass a telegraph post, then the length of the train is

Accepted Answer

Concept & Formula: First, find the speed of the train using the given distance and time. Once the uniform speed is known, use it along with the time taken to pass the pole to find the train's length. Speed = \frac{Distance}{Time} Length of Train = Speed (m/s) × Time to cross post (s) Step-by-Step Solution: • Given for Speed: Distance = 12 km = 12000 m. Time = 10 minutes = 10 × 60 = 600 seconds. • Calculate Speed: Speed = (12000/600) = 20 m/s. • Given for Crossing: Time to pass post = 6 seconds. • Calculate Length: Length = 20 m/s × 6 s = 120 m. Exam Strategy & Shortcut: 12 km in 10 minutes is equivalent to 72 km in 60 minutes (72 km/hr). We know 72 km/hr is exactly 20 m/s (4 × 18 -> 4 × 5). Multiplying 20 m/s by 6 seconds immediately yields 120 m. Common Pitfall: Mixing units. If you use 12 km and 10 minutes directly, you might end up with speed in km/min and apply it incorrectly to the 6 seconds. Always standardize to meters and seconds. Final Answer: Therefore, the correct answer is 120 m.

Question 2

A train 240 m long passed a pole in 24 seconds. How long will it take to pass a platform 650 m long?

Accepted Answer

Concept & Formula: When a train crosses a stationary pole, the distance covered is equal to its own length. This helps us find its speed. When crossing a platform, the total distance covered is the sum of the train's length and the platform's length. Speed = \frac{Length of Train}{Time to cross pole} Time to cross platform = \frac{Length of Train + Length of Platform}{Speed} Step-by-Step Solution: • Find Speed: The train covers 240 m in 24 seconds to cross the pole. • Speed = (240/24) = 10 m/s. • Find Total Distance: To cross the platform, distance = 240 m (train) + 650 m (platform) = 890 m. • Calculate Time: Time = \frac{Total Distance}{Speed} = (890/10) = 89 seconds. Exam Strategy & Shortcut: Observe the proportionality. The train covers 240 m in 24 seconds, meaning it takes exactly 1 second for every 10 meters. To cover a total distance of 890 meters (240 + 650), it will take 890 / 10 = 89 seconds. Common Pitfall: Calculating the time for the platform alone (650 / 10 = 65 seconds) without accounting for the train's own length that must also clear the platform. Final Answer: Therefore, the correct answer is 89 sec.

Question 3

A train 800 metres long is running at a speed of 78 km/hr. If it crosses a tunnel in 1 minute, then the length of the tunnel (in metres) is :

Accepted Answer

Concept & Formula: Crossing a tunnel is mathematically identical to crossing a platform. The total distance covered is the train's length plus the tunnel's length. Length of Tunnel = (Speed × Time) - Length of Train Step-by-Step Solution: • Given: Train length = 800 m, Speed = 78 km/hr, Time = 1 minute = 60 seconds. • Convert Speed: 78 × (5/18) = (390/18) = (65/3) m/s. • Calculate Total Distance: Distance = Speed × Time = (65/3) × 60. • Distance = 65 × 20 = 1300 meters. • Find Tunnel Length: 1300 m - 800 m = 500 meters. Exam Strategy & Shortcut: Keep terms in fractional form to allow for immediate simplification. Using (65/3) against the 60 seconds turns a potentially messy decimal multiplication into a clean 65 × 20. Common Pitfall: Forgetting to convert 1 minute into 60 seconds, and multiplying the speed by 1 instead. Final Answer: Therefore, the correct answer is 500.

Question 4

A 280-metre long train crosses a platform thrice its length in 50 seconds. What is the speed of the train in km/hr?

Accepted Answer

Concept & Formula: The total distance covered by the train to cross the platform is the sum of the train's length and the platform's length. Total Distance = L_{train} + L_{platform} Step-by-Step Solution: • Given: • Length of the train (L) = 280 m • Length of the platform = 3 × 280 = 840 m • Time taken (T) = 50 seconds • Calculation: • Total distance = 280 + 840 = 1120 m. • Speed in m/s = (1120/50) = (112/5) m/s. • Convert to km/hr = (112/5) × (18/5) = (2016/25). • (2016/25) = (2016 × 4/100) = (8064/100) = 80.64 km/hr. Exam Strategy & Shortcut: Instead of calculating 3 × 280 and adding 280, recognize that the total length is simply 4 times the train's length (4 × 280 = 1120). When dividing by 25, multiply the numerator by 4 and place the decimal two spots to the left. Common Pitfall: Students might calculate the platform's length as 3 × 280 but forget to add the train's original length when determining the total distance covered. Final Answer: Therefore, the correct answer is 80.64.

Question 5

A train, 150 m long, takes 30 seconds to cross a bridge 500 m long. How much time will the train take to cross a platform 370 m long?

Accepted Answer

Concept & Formula: When a train crosses an object with length, the total distance covered is the sum of the train's length and the object's length. First, calculate the speed of the train from the bridge scenario, and use it for the platform scenario. Speed = \frac{Total Distance}{Time} Step-by-Step Solution: • Given: • Length of train = 150 m. • Length of bridge = 500 m. • Time to cross bridge = 30 seconds. • Length of platform = 370 m. • Calculation: • Total distance for bridge = 150 + 500 = 650 m. • Speed of train = (650/30) = (65/3) m/s. • Total distance for platform = 150 + 370 = 520 m. • Time to cross platform = \frac{Distance}{Speed} = (520/\frac{65){3}} seconds. • Time = 520 × (3/65) = 8 × 3 = 24 seconds. Exam Strategy & Shortcut: Since speed is constant, the ratio of distances equals the ratio of times. Distance 1 = 650 m, Time 1 = 30 s. Distance 2 = 520 m, Time 2 = t. (650/30) = (520/t) \Rightarrow t = (520 × 30/650) = (52 × 3/6.5) = 24 seconds. Common Pitfall: A common mistake is forgetting to add the train's length to the platform's length when finding the second distance. Final Answer: Therefore, the correct answer is 24 sec.

Question 6

A train running at a certain speed takes 20 seconds to cross a signal post and 50 seconds to cross a bridge. Which of the following statements is correct about the length of the bridge?

Accepted Answer

Concept & Logic: Because the train travels at a constant speed, the distance covered is directly proportional to the time taken. The time to cross the bridge comprises the time to cross the train's own length plus the time to purely cross the bridge's length. Distance \propto Time Step-by-Step Solution: • Given: • Time to cross signal post (Train's length) = 20 seconds. • Time to cross bridge (Train's length + Bridge's length) = 50 seconds. • Calculation: • Time taken to cover *only* the bridge's length = Total time - Time for train's length. • Time for bridge's length = 50 - 20 = 30 seconds. • Ratio of Bridge's length to Train's length = Ratio of respective times. • Ratio = (30/20) = 1.5. • Thus, the bridge is 1.5 times the length of the train. Exam Strategy & Shortcut: Avoid assigning variables like x or y. Just look at the times: 20s for the train, leaving 30s for the bridge. (30/20) is exactly 1.5. Common Pitfall: A standard error is dividing 50 by 20 to get 2.5, which represents the ratio of the *total distance* to the train's length, not the *bridge's length* to the train's length. Final Answer: Therefore, the correct answer is 1.5 times the length of the train.

Question 7

A train travelling with a constant speed crosses a 96-metre long platform in 12 seconds and another 141-metre long platform in 15 seconds. The length of the train and its speed are

Accepted Answer

Concept & Formula: When a train crosses two platforms of different lengths, the difference in the time taken is entirely due to the difference in the lengths of the platforms. Speed = \frac{Difference in Platform Lengths}{Difference in Time} Step-by-Step Solution: • Given: • Platform 1: 96 m, Time 1: 12 seconds. • Platform 2: 141 m, Time 2: 15 seconds. • Calculation: • Difference in lengths = 141 - 96 = 45 m. • Difference in times = 15 - 12 = 3 seconds. • Speed of train = (45/3) = 15 m/s. • Convert speed to km/hr = 15 × (18/5) = 3 × 18 = 54 km/hr. • To find the length of the train (L): In the first scenario, total distance = L + 96. • L + 96 = Speed × Time = 15 × 12 = 180 m. • L = 180 - 96 = 84 m. Exam Strategy & Shortcut: Find the speed immediately: (141 - 96/15 - 12) = (45/3) = 15 m/s. Eliminate options with incorrect speeds (15 m/s = 54 km/hr, so options (b) and (d) are out). Then, simply calculate length: (15 × 12) - 96 = 84 m. Common Pitfall: Setting up two separate complex equations (L + 96 = 12S and L + 141 = 15S) and trying to solve them simultaneously wastes precious time compared to using the difference method. Final Answer: Therefore, the correct answer is 84 metres and 54 km/hr.

Question 8

A jogger running at 9 kmph alongside a railway track is 240 metres ahead of the engine of a 120 metre long train running at 45 kmph in the same direction. In how much time will the train pass the jogger?

Accepted Answer

Concept & Relative Speed: When two bodies move in the same direction, their relative speed is the difference between their individual speeds. To completely pass the jogger, the train must cover the initial distance gap plus its own entire length. Relative Speed = V_{train} - V_{jogger} Total Distance = Initial Gap + Length of Train Step-by-Step Solution: • Given: • Speed of jogger = 9 kmph. • Speed of train = 45 kmph. • Initial gap = 240 m. • Length of train = 120 m. • Calculation: • Relative speed = 45 - 9 = 36 kmph. • Convert to m/s: 36 × (5/18) = 10 m/s. • Total distance the train needs to cover relative to the jogger = 240 + 120 = 360 m. • Time taken = \frac{Total Distance}{Relative Speed} = (360/10) = 36 seconds. Exam Strategy & Shortcut: Instantly convert the relative speed of 36 kmph to 10 m/s. Add the gap and the train's length (360 m) in your head. Divide 360 by 10 to get 36 seconds immediately. Common Pitfall: A very common mistake is only calculating the time required to cover the 240m gap, forgetting that the train must also advance its own 120m length past the jogger to completely "pass" him. Final Answer: Therefore, the correct answer is 36 sec.

Question 9

A train 110 metres long is running with a speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?

Accepted Answer

Concept & Formula: When two bodies move in opposite directions, their relative speed is the sum of their individual speeds. Since a man is considered a point object, the distance to be covered is simply the length of the train. Relative Speed = V_{train} + V_{man} Step-by-Step Solution: • Given: • Length of train = 110 m. • Speed of train = 60 kmph. • Speed of man = 6 kmph. • Calculation: • Relative speed = 60 + 6 = 66 kmph. • Convert to m/s: 66 × (5/18) = (330/18) = (55/3) m/s. • Total distance = 110 m. • Time taken = \frac{Distance}{Relative Speed} = (110/\frac{55){3}} seconds. • Time = 110 × (3/55) = 2 × 3 = 6 seconds. Exam Strategy & Shortcut: Instead of fully simplifying (330/18), you can write the time equation as (110 × 18/330). The 110 goes into 330 exactly 3 times, leaving (18/3) = 6 seconds. Common Pitfall: Students often subtract the speeds by mistake out of habit. Always look for the keyword "opposite" to remind yourself to add the speeds together. Final Answer: Therefore, the correct answer is 6 sec.

Question 10

Two trains are moving in opposite directions @ 60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. The time taken by the slower train to cross the faster train in seconds is

Accepted Answer

Concept & Logic: When objects move in opposite directions, you add their speeds to find the relative speed. Since the lengths are already given in kilometers and speeds in km/hr, it is fastest to calculate the time in hours first, then convert directly to seconds. Relative Speed = S_1 + S_2 Step-by-Step Solution: • Given: • Speed of train 1 = 60 km/hr. • Speed of train 2 = 90 km/hr. • Length of train 1 = 1.10 km. • Length of train 2 = 0.9 km. • Calculation: • Relative speed = 60 + 90 = 150 km/hr. • Total distance = 1.10 + 0.9 = 2.0 km. • Time in hours = \frac{Distance}{Relative Speed} = (2/150) hours. • Convert time to seconds: (2/150) × 3600 = (2 × 360/15) = (720/15). • 720 \div 15 = 48 seconds. Exam Strategy & Shortcut: Keep units consistent to save time. Total distance is exactly 2 km. Relative speed is 150 km/hr. Time is (2/150) hr. Multiply by 3600 to get seconds: (7200/150) = (720/15) = 48. No need to convert km/hr to m/s or km to meters! Common Pitfall: A time-wasting pitfall is converting all km values to meters and km/hr to m/s right at the start. While it works, it introduces unnecessary arithmetic steps that increase the chance of calculation errors. Final Answer: Therefore, the correct answer is 48.

Question 11

Two trains are coming from opposite directions with speeds of 75 km/hr and 100 km/hr on two parallel tracks. At some moment the distance between them is 100 km. After T hours, distance between them is again 100 km. T is equal to (I.A.M., 2007)

Accepted Answer

Concept & Relative Speed: When two bodies move in opposite directions, their relative speed is the sum of their individual speeds. To be 100 km apart again, the trains must first meet (covering the initial 100 km separation) and then separate by another 100 km. Relative\ Speed = S_1 + S_2 Total\ Distance = D_{initial} + D_{final} Step-by-Step Solution: 1. Given: Speeds are 75 km/hr and 100 km/hr. Initial distance = 100 km, Final required distance = 100 km. 2. Calculate the relative speed since they are moving in opposite directions: 75 + 100 = 175 km/hr. 3. Determine the total relative distance they need to cover. They must cover the 100 km between them to cross paths, plus another 100 km to move apart again. Total distance = 100 + 100 = 200 km. 4. Calculate the time (T) taken to cover this relative distance: Time = (Distance/Relative\ Speed) T = (200/175) = (8/7) = 1 (1/7)\ hours Exam Strategy & Shortcut: Instead of modeling the exact meeting point, always use relative speed for these types of questions. Add the initial and final separation distances, and divide by the sum of speeds. 200 / 175 = 8/7. Common Pitfall: A common mistake is assuming the trains only need to cover 100 km total, forgetting that they must first cross the initial 100 km gap before creating a new 100 km gap. Final Answer: Therefore, the correct answer is 1 (1/7) hr.

Question 12

A train 75 m long overtook a person who was walking at the rate of 6 km/hr in the same direction and passed him in $7 \frac{1}{2}$ seconds. Subsequently, it overtook a second person and passed him in $6 \frac{3}{4}$ seconds. At what rate was the sec…

Accepted Answer

Concept & Successive Relative Speeds: Use the first overtaking scenario to find the train's absolute speed. Then, use that absolute speed in the second scenario to find the unknown speed of the second person. Speed_{relative} = \frac{Length_{train}}{Time} Step-by-Step Solution: 1. Analyze First Person: - Train length = 75 m, Time = 7.5 s. - Relative speed 1 = 75 / 7.5 = 10 m/s. - Convert to km/hr: 10 × (18/5) = 36 km/hr. - Since moving in the same direction, Speed_{train} - 6 = 36 \implies Speed_{train} = 42 km/hr. 2. Analyze Second Person: - Time = 6 (3/4) seconds = 27/4 s = 6.75 s. - Relative speed 2 = (75/27/4) = (300/27) = (100/9) m/s. - Convert to km/hr: (100/9) × (18/5) = 40 km/hr. - Since overtaking implies same direction, Speed_{train} - Speed_{person2} = 40. 3. Solve for Second Person's Speed: - 42 - Speed_{person2} = 40 - Speed_{person2} = 2 km/hr. Exam Strategy & Shortcut: Find both relative speeds in m/s first, convert to kmph, and track differences. Rel Speed 1: 10 m/s = 36 km/hr. Rel Speed 2: 100/9 m/s = 40 km/hr. Difference in relative speeds (40 - 36 = 4) must match the difference in the persons' speeds, but inversely. If relative speed increased by 4, the second person is moving 4 km/hr slower than the first. 6 - 4 = 2 km/hr. Common Pitfall: Miscalculating fractions when dividing by 6 (3/4). Always convert mixed fractions to improper fractions (27/4) before substituting into division operations. Final Answer: Therefore, the correct answer is 2 km/hr.

Question 13

Two trains are running in opposite directions with the same speed. If the length of each train is 120 metres and they cross each other in 12 seconds, then the speed of each train (in km/hr) is (S.S.C., 2003)

Accepted Answer

Concept & Equal Speeds in Opposite Directions: When two identical objects move toward each other at the same speed, the relative speed is twice the speed of one object. The distance covered to cross is the sum of their lengths. Relative\ Speed = 2v Total\ Distance = 2L Step-by-Step Solution: 1. Given: Length of each train = 120 m. Total distance = 120 + 120 = 240 m. Time = 12 s. Let the speed of each train be v m/s. 2. The relative speed since they are in opposite directions is v + v = 2v. 3. Calculate relative speed using distance and time: Relative\ Speed = (Distance/Time) = (240/12) = 20\ m/s 4. Set the equations equal to find the speed of one train: 2v = 20 \implies v = 10\ m/s 5. Convert the speed from m/s to km/hr: 10 × (18/5) = 36\ km/hr Exam Strategy & Shortcut: Total distance is 240m. Time is 12s. Combined speed = 240/12 = 20 m/s. Since speeds are equal, one train's speed is 10 m/s. Multiply by 18/5 to get 36 km/hr instantly. Common Pitfall: Stopping the calculation at the relative speed (20 m/s or 72 km/hr) and choosing that as the answer, forgetting that it represents the *combined* speed of both trains. Final Answer: Therefore, the correct answer is 36.

Question 14

A 180-metre long train crosses another 270-metre long train running in the opposite direction in 10.8 seconds. If the speed of the first train is 60 kmph, what is the speed of the second train in kmph? (Bank P.O., 2010)

Accepted Answer

Concept & Converting Combined Distance: To find an unknown speed in an opposite-direction crossing, first determine the total combined speed in m/s, convert it to km/hr, and then subtract the known speed. Relative\ Speed = (L_1 + L_2/Time) Step-by-Step Solution: 1. Given: L_1 = 180 m, L_2 = 270 m. Total distance = 180 + 270 = 450 m. Time = 10.8 s. S_1 = 60 kmph. Let S_2 be the speed of the second train in kmph. 2. Calculate the relative speed in m/s: Relative\ Speed = (450/10.8) = (4500/108)\ m/s 3. Convert this relative speed to km/hr: (4500/108) × (18/5) = (900/108) × 18 = (900/6) = 150\ km/hr 4. Since they move in opposite directions, the relative speed is the sum of their individual speeds: S_1 + S_2 = 150 60 + S_2 = 150 5. Solve for S_2: S_2 = 150 - 60 = 90\ km/hr Exam Strategy & Shortcut: Distance = 450m. Time = 10.8s. Speed in km/hr = (450/10.8) × (18/5) = 150 km/hr. Since opposite direction means speeds are added, the second train's speed is 150 - 60 = 90 km/hr. Common Pitfall: Struggling with the decimal division (450/10.8). Always clear decimals by multiplying numerator and denominator by 10 before simplifying, and apply the 18/5 conversion early to cancel out awkward denominators like 108 (since 18 × 6 = 108). Final Answer: Therefore, the correct answer is 90.

Question 15

Two trains are running in opposite directions with the same speed. If the length of each train is 120 metres and they cross each other in 12 seconds, then the speed of each train (in km/hr) is (S.S.C., 2003)

Accepted Answer

Concept & Equal Speeds in Opposite Directions: When two identical objects move toward each other at the same speed, the relative speed is twice the speed of one object. The distance covered to cross is the sum of their lengths. Relative\ Speed = 2v Total\ Distance = 2L Step-by-Step Solution: 1. Given: Length of each train = 120 m. Total distance = 120 + 120 = 240 m. Time = 12 s. Let the speed of each train be v m/s. 2. The relative speed since they are in opposite directions is v + v = 2v. 3. Calculate relative speed using distance and time: Relative\ Speed = (Distance/Time) = (240/12) = 20\ m/s 4. Set the equations equal to find the speed of one train: 2v = 20 \implies v = 10\ m/s 5. Convert the speed from m/s to km/hr: 10 × (18/5) = 36\ km/hr Exam Strategy & Shortcut: Total distance is 240m. Time is 12s. Combined speed = 240/12 = 20 m/s. Since speeds are equal, one train's speed is 10 m/s. Multiply by 18/5 to get 36 km/hr instantly. Common Pitfall: Stopping the calculation at the relative speed (20 m/s or 72 km/hr) and choosing that as the answer, forgetting that it represents the *combined* speed of both trains. Final Answer: Therefore, the correct answer is 36.

Question 16

A 180-metre long train crosses another 270-metre long train running in the opposite direction in 10.8 seconds. If the speed of the first train is 60 kmph, what is the speed of the second train in kmph? (Bank P.O., 2010)

Accepted Answer

Concept & Converting Combined Distance: To find an unknown speed in an opposite-direction crossing, first determine the total combined speed in m/s, convert it to km/hr, and then subtract the known speed. Relative\ Speed = (L_1 + L_2/Time) Step-by-Step Solution: 1. Given: L_1 = 180 m, L_2 = 270 m. Total distance = 180 + 270 = 450 m. Time = 10.8 s. S_1 = 60 kmph. Let S_2 be the speed of the second train in kmph. 2. Calculate the relative speed in m/s: Relative\ Speed = (450/10.8) = (4500/108)\ m/s 3. Convert this relative speed to km/hr: (4500/108) × (18/5) = (900/108) × 18 = (900/6) = 150\ km/hr 4. Since they move in opposite directions, the relative speed is the sum of their individual speeds: S_1 + S_2 = 150 60 + S_2 = 150 5. Solve for S_2: S_2 = 150 - 60 = 90\ km/hr Exam Strategy & Shortcut: Distance = 450m. Time = 10.8s. Speed in km/hr = (450/10.8) × (18/5) = 150 km/hr. Since opposite direction means speeds are added, the second train's speed is 150 - 60 = 90 km/hr. Common Pitfall: Struggling with the decimal division (450/10.8). Always clear decimals by multiplying numerator and denominator by 10 before simplifying, and apply the 18/5 conversion early to cancel out awkward denominators like 108 (since 18 × 6 = 108). Final Answer: Therefore, the correct answer is 90.

Question 17

Two trains, each 100 m long, moving in opposite directions, cross each other in 8 seconds. If one is moving twice as fast the other, then the speed of the faster train is :

Accepted Answer

Concept & Relative Speed: When two trains move in opposite directions, their relative speed is the sum of their individual speeds. They must cover a distance equal to the sum of their lengths to cross each other. Relative Speed = \frac{Total Length}{Time} Step-by-Step Solution: • Total distance to cover = Sum of lengths = 100 + 100 = 200 meters. • Time taken to cross = 8 seconds. • Relative speed in m/s = (200/8) = 25 m/s. • Convert relative speed to km/hr: 25 × (18/5) = 90 km/hr. • Let the speed of the slower train be x km/hr. The faster train is 2x km/hr. • Relative speed = x + 2x = 3x. • Equating speeds: 3x = 90 \Rightarrow x = 30 km/hr. • Speed of the faster train = 2x = 60 km/hr. Exam Strategy & Shortcut: Calculate relative speed directly in m/s, assign ratios (1:2 means 3 parts total), find the value of one part (25 / 3 m/s), and then calculate the faster train's speed before converting: 2 × (25/3) × (18/5) = 60 km/hr. Common Pitfall: Forgetting to convert the relative speed from m/s to km/hr before finding the individual speeds, leading to incorrect choices. Final Answer: Therefore, the correct answer is 60 km/hr.

Question 18

A 150 m long train crosses a milestone in 15 seconds and a train of same length coming from the opposite direction in 12 seconds. The speed of the othe train is (R.R.B., 2006)

Accepted Answer

Concept & Distance: A train crossing a stationary object (like a milestone) covers a distance equal to its own length. When crossing another moving train, the distance is the sum of both lengths. Speed = \frac{Length}{Time} Step-by-Step Solution: • Length of the first train = 150 m. Time to cross milestone = 15 s. • Speed of the first train = (150/15) = 10 m/s. • Length of the second train = 150 m (since it has the same length). • Total distance when crossing each other = 150 + 150 = 300 m. • Time taken to cross each other = 12 s. • Relative speed = (300/12) = 25 m/s. • Since they move in opposite directions, Relative Speed = Speed 1 + Speed 2. • 10 + Speed 2 = 25 \Rightarrow Speed 2 = 15 m/s. • Convert to kmph: 15 × (18/5) = 54 kmph. Exam Strategy & Shortcut: You can calculate relative speed 25 m/s and subtract the first train's speed (10 m/s) to get 15 m/s. Multiply by 3.6 for a rapid m/s to km/h conversion: 15 × 3.6 = 54 kmph. Common Pitfall: Assuming the second train crosses a point in 12 seconds instead of crossing the first train. Always read whether it crosses a point or an object with length. Final Answer: Therefore, the correct answer is 54 kmph.

Question 19

A train overtakes two persons who are walking in the same direction in which the train is going, at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train is :

Accepted Answer

Concept & Equations of Motion: When a train overtakes people walking in the same direction, the relative speed is the train's speed minus the person's speed. The length of the train is constant, so we can equate the distance formulas for both scenarios. Length = (Speed_{train} - Speed_{person}) × Time Step-by-Step Solution: • Let the speed of the train be v kmph and its length be L meters. • For the first person: Speed is 2 kmph, time is 9 seconds. • Relative speed = (v - 2) kmph. Length L = (v - 2) × (5/18) × 9. • For the second person: Speed is 4 kmph, time is 10 seconds. • Relative speed = (v - 4) kmph. Length L = (v - 4) × (5/18) × 10. • Equate both expressions for L: (v - 2) × 9 = (v - 4) × 10. (The (5/18) factor cancels out). • Solve for v: 9v - 18 = 10v - 40 \Rightarrow v = 22 kmph. • Substitute v back to find L: L = (22 - 2) × (5/18) × 9 = 20 × (5/2) = 50 meters. Exam Strategy & Shortcut: Use the cross-multiplication formula to find the train's speed directly: v = ((v_2 × t_2) - (v_1 × t_1)/t_2 - t_1). Here, v = ((4 × 10) - (2 × 9)/10 - 9) = (40 - 18/1) = 22 kmph. Then, calculate length: L = (22 - 2) × (5/18) × 9 = 50 m. Common Pitfall: Forgetting to convert the kmph speed to m/s when calculating the final length L. The conversion (5/18) cancels out while finding speed, but is strictly required for the length calculation. Final Answer: Therefore, the correct answer is 50 m.

Question 20

A train overtakes two persons walking along a railway track. The first one walks at 4.5 km/hr. The other one walks at 5.4 km/hr. The train needs 8.4 and 8.5 seconds respectively to overtake them. What is the speed of the train if both the persons ar…

Accepted Answer

Concept & Relative Speed: The length of the train remains constant when it overtakes two different moving objects. By setting the distance expressions (relative speed multiplied by time) equal to each other, we can solve for the unknown speed of the train. Distance = (v - u) × t Step-by-Step Solution: • Let the speed of the train be v km/hr. • Relative speed with respect to the first person = (v - 4.5) km/hr. Time taken t_1 = 8.4 seconds. • Length of train L = (v - 4.5) × (5/18) × 8.4. • Relative speed with respect to the second person = (v - 5.4) km/hr. Time taken t_2 = 8.5 seconds. • Length of train L = (v - 5.4) × (5/18) × 8.5. • Equate the two length equations: (v - 4.5) × 8.4 = (v - 5.4) × 8.5. (The (5/18) constant cancels out). • Expand the terms: 8.4v - 37.8 = 8.5v - 45.9. • Rearrange to solve for v: 45.9 - 37.8 = 8.5v - 8.4v. • 8.1 = 0.1v \Rightarrow v = (8.1/0.1) = 81 km/hr. Exam Strategy & Shortcut: To quickly find the speed without full algebra, use the difference formula: v = (v_2 × t_2 - v_1 × t_1/t_2 - t_1). Calculate (5.4 × 8.5) - (4.5 × 8.4) divided by (8.5 - 8.4). 45.9 - 37.8 = 8.1. Divide by 0.1 to get 81 km/hr. Common Pitfall: Making arithmetic errors when multiplying decimals like 5.4 × 8.5 or 4.5 × 8.4. Careful decimal placement is critical here. Final Answer: Therefore, the correct answer is 81 km/hr.

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