Problem 4 - Technical Mathematics with Calculus

Question

A freight train leaves \(A\) for \(B, 175\) miles away, and travels at the rate of 31.5 mi/h. After 1.50 hours, a train leaves \(B\) for \(A,\) traveling at 21.5 mi/h. How many miles from \(B\) will they meet?

Step-by-Step Solution

Answer

The trains will meet 86 miles from B.

Step 1: Calculate distance covered by Freight Train before Second Train starts

Multiply the speed of the freight train by the time before the second train starts to find out how far it has traveled. Distance = Speed × Time, therefore Distance = 31.5 mi/h × 1.50 h.

Step 2: Calculate the effective speed at which the gap is closing

After both trains are traveling towards each other, the gap between them closes at the sum of their speeds. Effective Speed = Speed of first train + Speed of second train, so Effective Speed = 31.5 mi/h + 21.5 mi/h.

Step 3: Calculate the remaining distance to be covered

Subtract the distance covered by the freight train from the total distance to find the remaining distance. Remaining Distance = Total Distance - Distance covered by the freight train.

Step 4: Calculate the time it will take for the trains to meet

Divide the remaining distance by the effective speed to find the time it will take for the trains to meet. Time = Remaining Distance / Effective Speed.

Step 5: Calculate the distance from B when the trains meet

Multiply the speed of the second train by the time calculated in Step 4 to find the distance from B when the trains meet. Distance from B = Speed of second train × Time.

Key Concepts

Understanding Relative Speed

Relative speed comes into play when two objects are moving in relation to each other. It's especially important in rate time distance problems, like the one involving two trains starting from points A and B heading towards each other.

In such scenarios, the relative speed is the sum of the individual speeds if the objects are moving towards each other, or the difference if they are moving in the same direction. Imagine you're walking down a corridor towards a friend who's walking towards you; your relative speed is how quickly the distance between you is changing.

For the trains in our problem, their relative speed is simply 31.5 mi/h + 21.5 mi/h, because they are moving toward each other. This concept is crucial as it helps us determine how quickly the distance between them diminishes.

The Principle of Uniform Motion

Uniform motion refers to moving at a constant speed or velocity, without acceleration or deceleration. In our train problem, both trains are assumed to be traveling at uniform motion, which means their speeds remain constant during their respective journeys.

Understanding uniform motion is important because it allows us to predict where and when the two trains will meet. With constant speeds, the distance each train travels over a given time can be easily calculated using the formula: Distance = Speed × Time.

Knowing the distances involved, we can then find the exact point where the trains converge by using this principle. Uniform motion simplifies the mathematical process and allows us to use straightforward algebra to solve for the unknowns.

Solving With Algebraic Equations

Algebraic equations are the backbone of solving rate time distance problems effectively. By setting up an equation, we create a mathematical representation of the scenario at hand.

In our exercise, by multiplying the speed of the first train by the time it traveled before the second train started, we acquire the first distance. We then calculate the remaining distance and use it to find the time until they meet.

The beauty of algebra lies in its ability to transform real-world problems into solvable equations. By understanding how to manipulate these equations, you can solve for the unknown variable—such as the distance from point B when the trains meet—using simple but powerful algebraic manipulation.

Train Motion Problems

Train motion problems are a classic subset of distance, rate, and time questions encountered in mathematics. They often involve trains moving towards or away from each other or a stationary object.

These problems are practical, as they reflect the real dynamics of trains in motion, making them an excellent way for students to apply mathematical concepts to lifelike scenarios. By understanding the principles of relative speed and uniform motion and using algebraic equations, students can tackle these problems with confidence.

Remember, to solve these, draw a simple diagram to visualize the problem, set up equations for the distances traveled, apply the concept of relative speed, and finally, use algebra to find the unknowns. This approach not only gives the answer but also strengthens problem-solving skills in a tangible, real-world context.