Solving Two-Object Motion Problems

[Illicitsky]

Homework Statement

1. Two people leave from two towns that are 195 miles apart at the same time and travel along the same road toward eah other. The first person drives 5 miles slower than the second person. If they meet in 3 hours, at what rate of speed did each travel?

Homework Equations

d= rt

The Attempt at a Solution

Simple motion problems are easy. But HOW do we figure out problems when two objects are going down the same road/path, with X miles in between or even 'overtake' the other?

No idea where to start. Tried to set up d=rt chart by there's too many variables in the problem (the 3 hours thing, then 5 miles slower?)

 

[ehild]

There are two towns, A and B, D distance apart. There are two people: the first starting from A with speed v_a, the other starts from B with speed v_b=v_a-3.

Denote the distance of the people from A by x_a and x_b. At the beginning, x_a=0, x_b=D

After t time elapsed, the first man is x_a=v_at distance from A. The other man is x_b=D-v_bt distance from A.

When they meet, they both are at the same distance from A: x_a=x_b.

ehild

 

[kerimek]

I can provide a few steps to help solve this type of problem. First, we need to identify the known information and what we are trying to solve for. In this case, we know the distance between the two towns (195 miles), the time it took for them to meet (3 hours), and the fact that one person is driving 5 miles slower than the other. We are trying to find the rate of speed for each person.

Next, we can use the equation d=rt to create two equations, one for each person's distance traveled. Let's call the rate of speed for the first person r and the rate of speed for the second person r+5 (since they are traveling 5 miles faster).

Person 1's distance traveled: d=rt
Person 2's distance traveled: d=(r+5)t

Since they are traveling towards each other, their combined distance will equal the distance between the two towns (195 miles). So we can set up an equation with this information:

195 = (r+5)t + rt

Now we have two equations with two unknowns (r and t). We can solve for one of the variables and then use that value to solve for the other variable.

Let's start by solving for t:

195 = (r+5)t + rt
195 = rt + 5t + rt
195 = 2rt + 5t
195 = t(2r+5)
t = 195/(2r+5)

Now we can substitute this value for t into either one of the original equations to solve for r. Let's use the first equation:

Person 1's distance traveled: d=rt
195 = r(195/(2r+5))
195 = 195r/(2r+5)
2r+5 = 195r/195
2r+5 = r
5 = r

So we have found that the first person was traveling at a rate of 5 miles per hour and the second person was traveling at a rate of 10 miles per hour (since they were going 5 miles per hour faster).

In summary:

Person 1: r=5 miles per hour
Person 2: r+5 = 5+5 = 10 miles per hour

I hope this helps! Remember, when solving two-object motion problems, it's important to identify the known information, set up equations using the