Question

Spacecraft \(A\) is over Houston at noon on a certain day and traveling at a rate of \(275 \mathrm{km} / \mathrm{h}\). Spacecraft \(B,\) attempting to overtake and dock with \(A\) is over Houston at 1: 15 P.M. and is traveling in the same direction as \(A\), at \(444 \mathrm{km} / \mathrm{h} .\) At what time will \(B\) overtake \(A ?\) At what distance from Houston?

Short answer

Spacecraft B will overtake spacecraft A at 2:39 P.M., 583.5 km from Houston.

Step-by-step solution

Calculate the lead of spacecraft A

Determine how far spacecraft A has traveled by 1:15 P.M. Since spacecraft A leaves at noon and travels at 275 km/h, we calculate the distance using the formula distance = speed × time. Spacecraft A has a 1 hour and 15 minute lead on spacecraft B, which is 1.25 hours. The distance traveled by spacecraft A by 1:15 P.M. is then 275 km/h × 1.25 h.

Determine the speed difference

Find the relative speed at which spacecraft B is approaching spacecraft A. This can be found by subtracting the speed of spacecraft A from the speed of spacecraft B. The relative speed is 444 km/h - 275 km/h.

Calculate the time required for spacecraft B to catch up

Using the lead distance and the relative speed, calculate the time it takes for spacecraft B to catch up with spacecraft A. The time is calculated by dividing the lead distance by the relative speed.

Find the overtaking time

Add the time calculated in Step 3 to the time spacecraft B started (1:15 P.M.) to find the exact time of the day when spacecraft B will overtake spacecraft A.

Calculate the overtaking distance from Houston

Multiply the speed of spacecraft B by the time taken to overtake spacecraft A (found in Step 3) to get the distance from Houston where the overtaking occurs.

Key concepts

Understanding the Distance = Speed × Time Formula

One of the fundamental principles of motion states that the distance an object travels is equal to its speed multiplied by the time it's been moving. This equation is deceptively simple but exceedingly powerful. To visualize its application, imagine a runner moving at a constant speed of 10 kilometers per hour. If they run for 2 hours, the distance they cover would be \[\begin{equation} 10 \text{ km/h} \times 2 \text{ h} = 20 \text{ km}. \end{equation}\]This relationship is directly used to calculate how far an object has gone, how long it took to travel a certain distance, or how fast it was moving.In the exercise with spacecraft A, the application of this formula provided us with the distance the spacecraft traveled from Houston until the point when spacecraft B started its pursuit. These initial conditions set the stage for the rest of the problem-solving process.

Calculating Lead Distance

The concept of lead distance is central to problems involving pursuit and overtaking scenarios. It is essentially the head start or advantage one moving object has over another. We calculate it by determining how far the leading object has traveled before the chasing object begins its pursuit.For instance, if a car departs one hour before another from the same location and drives at a steady speed of 60 km/h, its lead distance after an hour would be \[\begin{equation} 60 \text{ km/h} \times 1 \text{ h} = 60 \text{ km}. \end{equation}\]Understanding this lead distance is pivotal because it tells us how much ground the second car needs to cover to catch up. In our spacecraft scenario, the lead distance is the space between where spacecraft A is at 1:15 P.M. and Houston, the starting point for both spacecraft.

Relative Motion in One Dimension

When objects are in motion along the same straight path, we describe their movement in relation to each other using the principles of relative motion in one dimension. This involves understanding how the speed of one object compares to the speed of another.If, for example, two cars are driving on the same road, where Car A travels at 80 km/h and Car B at 100 km/h, then Car B's speed relative to Car A is \[\begin{equation} 100 \text{ km/h} - 80 \text{ km/h} = 20 \text{ km/h}. \end{equation}\]This relative speed is how fast Car B is gaining on or moving away from Car A. In the context of our spacecrafts, we calculated the relative speed of spacecraft B to spacecraft A to determine how quickly B could catch up to A.

Determining Overtaking Time

Overtaking time is the duration it takes a moving object to catch up to and pass another moving object. To find this, we need the lead distance and the relative speed. The time for one to overtake the other is the lead distance divided by the relative speed.Let's consider a cyclist who has a 30-minute head start on another, and the second cyclist is cycling 5 km/h faster than the first. If the first cyclist's speed is 15 km/h, the lead distance would be \[\begin{equation} 15 \text{ km/h} \times 0.5 \text{ h} = 7.5 \text{ km}. \end{equation}\]Given the second cyclist's relative speed of \[\begin{equation} 15 \text{ km/h} + 5 \text{ km/h} = 20 \text{ km/h}, \end{equation}\]the overtaking time in hours is \[\begin{equation} \frac{7.5 \text{ km}}{20 \text{ km/h}} = 0.375 \text{ h}. \end{equation}\]In our spacecraft example, after finding the spacecrafts’ relative speed, we use it to calculate how long spacecraft B will take to overtake spacecraft A, then determine the exact time of day this event will occur and the distance from the starting point.