# Question Video: Using Relative Velocity to Find the Length of a Train given the Time Taken by a Moving Object to Pass It Mathematics

A helicopter flew in a straight line at 234 km/h above a train moving in the same direction. It took the helicopter 21 seconds to travel the length of the train. Following this, the pilot halved the helicopter’s speed. Given that it took the train 14 seconds to pass the helicopter traveling at this speed, find the length of the train in meters.

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### Video Transcript

A helicopter flew in a straight line at 234 kilometers per hour above a train moving in the same direction. It took the helicopter 21 seconds to travel the length of the train. Following this, the pilot halved the helicopter’s speed. Given that it took the train 14 seconds to pass the helicopter traveling at this speed, find the length of the train in meters.

Let’s begin by sketching the scenario in this question. We have a helicopter flying in the same direction as a train. The helicopter is initially flying at 234 kilometers per hour. And we are told it takes 21 seconds for the helicopter to travel the length of the train. Using our speed-distance-time triangle, we know that the distance traveled is equal to the speed multiplied by the time. In this question, the speed will be the relative velocity, where the velocity of 𝐴 relative to 𝐵 is equal to the velocity of 𝐴 minus the velocity of 𝐵. This is true for any two bodies moving along the same one-dimensional axis.

In this question, the velocity of the helicopter relative to the train will be equal to the velocity of the helicopter minus the velocity of the train. We have an issue here with our units as the velocity is given in kilometers per hour, but the time is given in seconds. We know that there are 1000 meters in a kilometer and 3600 seconds in an hour. To convert from kilometers per hour to meters per second, we multiply by 1000 and divide by 3600. This is the same as dividing by 3.6. 234 divided by 3.6 is equal to 65. Therefore, the velocity of the helicopter is 65 meters per second. This means that the velocity of the helicopter relative to the train is equal to 65 minus 𝑉 sub 𝑇. Since it took the helicopter 21 seconds to travel the length of the train, the distance traveled 𝑑 is equal to 65 minus 𝑉 sub 𝑇 multiplied by 21.

As there are two unknowns, we cannot solve this equation. We will call it equation one and now consider the second part of the question. We are told that the pilot of the helicopter halved its speed. As half of 65 is 32.5, the velocity of the helicopter is now 32.5 meters per second. Clearing some space and noticing that now it took the train 14 seconds to pass the helicopter, the velocity of the train relative to the helicopter is now equal to the velocity of the train minus the velocity of the helicopter. This is equal to 𝑉 sub 𝑇 minus 32.5. The distance traveled or the length of the train 𝑑 is this time equal to 𝑉 sub 𝑇 minus 32.5 multiplied by 14.

We now have two simultaneous equations that we can use to find the value of 𝑉 sub 𝑇 and hence the length of the train. Equating the right-hand side of both equations, we have 21 multiplied by 65 minus 𝑉 sub 𝑇 is equal to 14 multiplied by 𝑉 sub 𝑇 minus 32.5. We can divide through by seven and then distribute our parentheses, giving us 195 minus three 𝑉 sub 𝑇 is equal to two 𝑉 sub 𝑇 minus 65. Adding three 𝑉 sub 𝑇 and 65 to both sides, we have 260 is equal to five 𝑉 sub 𝑇. We can then divide through by five, giving us 𝑉 sub 𝑇 is equal to 52. The velocity of the train is equal to 52 meters per second.

We can now substitute this value into equation one or equation two. Substituting into equation one, we have 𝑑 is equal to 65 minus 52 multiplied by 21. This simplifies to 21 multiplied by 13, which is equal to 273. We can therefore conclude that the length of the train is 273 meters.