Cora is a passenger in a car traveling east along a country highway that is parallel to a railroad track and clear of any traffic. Her friend Liz is driving. They come up behind the last car of a train on the track. Initially, the train is moving a little slower than they are. For fun, Cora decides she is interested in finding out the total length of the train.
a. Describe what Cora should tell Liz about how to drive in order to find the speed of the train, and then measure the length of the train.
b. Say they find the train is moving 45 mph. Then it takes them ten minutes to pass the train (along its full length) after they speed up to 60 mph. How long is the train? For credit, please include an explanation of how you solved the problem.
a. To find the speed of the train and measure its length, Cora should advise Liz to maintain a constant speed and distance from the train. They should keep track of the time it takes for them to pass the train completely and record their own speed during this process.
b. Given that the train is moving at 45 mph and it takes them 10 minutes (or 10/60 = 1/6 hours) to pass the train after they accelerate to 60 mph, we can calculate the length of the train.
First, we need to determine the relative speed between the car and the train. Since the train is moving slower initially, the relative speed will be the difference in their speeds. Therefore, the relative speed is 60 mph - 45 mph = 15 mph.
Next, we can calculate the distance traveled by the car during the time it takes to pass the train. The distance traveled by the car is the relative speed multiplied by the time taken to pass the train.
Distance traveled by the car = Relative speed * Time taken to pass the train
Distance traveled by the car = 15 mph * 1/6 hours
Distance traveled by the car = 2.5 miles
Since the distance traveled by the car is equal to the length of the train, we can conclude that the train is 2.5 miles long.
a. In order to find the speed of the train and measure its length, Cora should tell Liz to do the following:
1. Maintain a constant speed while driving behind the train until they are parallel to it.
2. Once they are parallel to the train, make a note of their own speed. This is the relative speed between the car and the train.
3. Cora should observe a particular point on the train (such as a distinctive marking or an end carriage) and note the time it takes for that point to pass them completely.
4. Cora should also note the time it takes for the entire train to pass them.
By measuring the relative speed between the car and the train and the time it takes for a specific point and the entire train to pass, Cora and Liz can calculate the speed of the train and its length.
b. Given that the train is moving at 45 mph and it takes them 10 minutes to pass the train after accelerating to 60 mph, we can calculate the length of the train using the following steps:
1. Convert the time it takes to pass the train from minutes to hours. 10 minutes is one-sixth of an hour (10/60).
2. Calculate the distance they traveled to pass the train at a relative speed of 60 mph. Distance = Speed x Time. In this case, Distance = 60 mph x (1/6) hours.
3. Subtract the distance they traveled at a relative speed of 45 mph from the total distance traveled to pass the train. This difference will give us the length of the train.
Let's calculate the length of the train:
Distance traveled at 60 mph = 60 mph x (1/6) hours = 10 miles
Distance traveled at 45 mph = 45 mph x (1/6) hours = 7.5 miles
Length of the train = Distance traveled at 60 mph - Distance traveled at 45 mph = 10 miles - 7.5 miles = 2.5 miles
Therefore, the length of the train is 2.5 miles.