Two cars leave the beach at different times

Question

You are going on a trip to the beach with some friends in two cars and leaving at separate times. The first car leaves at 9AM and travels at a speed of 45mph. The second car leaves at 10:30 and travels at a speed of 55mph. The beach is 250 miles away. Will the second car catch up to the first car before arriving to the beach? why or why not? support your answer with work.

Answer 1

First, lets check what time each car will arrive at the beach. We will use the distance formula here:

distance = velocity * time

car 1:

250 miles = 45mph * t

t = 5.55 hrs = 5 hours 33 minutes

9 am + 5 hours 33 minutes = 2:33 pm

car 2:

250 miles = 55mph * t

t = 4.54 hrs = 4 hours 32 minutes

10:30 am + 4 hours 32 minutes = 3:02 pm

Car 1 started driving and arrived before car 2, so car 2 does not catch up to car 1 before arriving at the beach.

Answer 2

There are several ways you could do this problem, so what I'm showing you is just one way. An important thing to know before you can answer this question is that speed, or velocity, is equal to distance over time. That's why speeds are given in units of some distance/time, for example miles per hour, or miles/hour. It's also important that your units are consistent in these types of problems. Here we are given a distance in miles, and a speed in miles/hour, which is good because they both use miles.

Usually in these types of problems, you need a good reference point. For this one, let's say our starting point is actually at 10:30, when the second car starts their trip and the first car has been travelling for 1.5 hours. How do we know the first car has been travelling for 1.5 hours? Because it started at 9. 10:30 means 30 minutes has passed since 10, and since 30 minutes is half an hour, you could say it's 10.5. 10.5 - 9 = 1.5.

Back the main problem, at our reference point, 10:30, we need to know the distance travelled for both cars. For the second car, it's easy, they're just starting their trip, so their distance is 0.

For the first car, it's been travelling for 1.5 hours. To figure out the distance travelled, you will need to know an object's speed and the time it has been going that speed. Here the speed was given as 45 miles/hour. Remember that speed, or velocity, is equal to distance over time. Writing that in math terms could look like this:

v = d/t, where v = velocity (or speed)

d = distance travelled

t = time travelled

To solve for distance d, just multiply both sides by time t.

v * t = d/t * t

v * t = d

Using our numbers, we know their velocity is 45 miles/hour, the time travelled is 1.5 hours, so plugging that in gives us:

(45 miles/hour) * (1.5 hours) = 67.5 miles

So now we know that the first car's distance travelled is 67.5 miles.

Now that we know their starting positions using our reference point, let's write an equation that tells us both cars positions to see if they are ever equal. For this equation, we will use what we know about velocity and their starting positions. Writing this in math terms could look like this:

p_f = v * t + p_i, where p_f = final position, or the position we want to calculate

v = velocity

t = time travelled

p_i = initial position

For the second car, we know their velocity is 55 miles/hour and their starting position using our reference point is 0. So their position equation will look like this:

p_f = 55 * t + 0

= 55t

Now for the first car, we know their velocity is 45 miles/hour and their starting position using our reference point is 67.5 miles. So their position equation will look like this:

p_f = 45 * t + 67.5

= 45t + 67.5

Now that we can calculate both cars' positions, let's set these equations equal to each other and solve for the time makes their positions equal, or when the second car catches up to the first.

55t = 45t + 67.5

10t = 67.5

t = 6.75 hours

So after the second car starts their trip, it would take them 6.75 hours to catch up to the first. But we have to check to see if they will actually be travelling for 6.75 hours, or will their trip take less time? Remember, they're only travelling 250 miles to the beach. Using the velocity equation, we can solve for time t to figure out how long it will take to travel a distance going a certain speed.

v = d/t

v * t = d

t = d/v

Now that have our equation to solve for the time it takes to travel a certain distance going a certain speed, we can figure out how long it will take them to travel the 250 miles to the beach. Our distance is 250 miles, our velocity is 55 miles/hour, so using the equation gives us:

t = 250 miles/55 miles/hour

= 4.55 hours

So their trip will only take 4.55 hours, which is less than the 6.75 hours it would take to catch up to the first car. So no, the second car won't catch up to the first car.

If you wanted to see how far they would have to travel to catch up to the first car, plug the 6.75 hours it would take to catch up into the second car's position equation.

p_f = 55t

= 55 miles/hour (6.75 hours)

= 371.25 miles

Answer 3

One approach is to write the equations for how far the cars go as a function of time and set them equal to each other in order to find the time that they are at the same position (i.e. the fast car catches the slower car). You can then calculate the distance the two cars went by plugging in the time to either equation and compare it with 250 miles.

d₁ = 45t

d₂ = 55(t-1.5)

where t is in hours and d is in miles