Question

Two cars are traveling 40 and 50 miles per hour, respectively. If the second car starts out 5 miles behind the first car, how long will it take the second car to overtake the first car?

Short answer

The second car will overtake the first car in 30 minutes.

Step-by-step solution

Write the equation for the first car's distance covered

Let D1 be the distance covered by the first car. Let t be the time it will take the second car to catch up with the first car. The distance covered by the first car can be represented as: D1 = speed1 * t D1 = 40t

Write the equation for the second car's distance covered

Again, let D2 be the distance covered by the second car. The distance covered by the second car can be represented as: D2 = speed2 * t D2 = 50t

Write the equation that represents when the second car catches up with the first car

When the second car catches up with the first car, their distances will be equal. However, we must also account for the fact that the second car starts out 5 miles behind the first car. So, taking this into account, we can write: D2 = D1 + 5

Substitute the expressions for D1 and D2

Now, we substitute the expressions for D1 and D2 from Steps 1 and 2 into the equation from Step 3: 50t = 40t + 5

Solve the equation for t

To solve for the time t, we can first subtract 40t from both sides: 10t = 5 Now, we can divide both sides by 10: t = \(\frac{1}{2}\)

Calculate the time in hours

The time \(t\) we found is in hours. To find the time in hours, we can multiply the fraction by the number of minutes in an hour: Time = \(t\) * 60 minutes Time = \(\frac{1}{2}\) * 60 minutes = 30 minutes The second car will overtake the first car in 30 minutes.

Key concepts

Distance Formula

In the context of relative motion, the distance formula is a crucial tool. It helps in figuring out how far two objects move relative to each other over time. In this scenario, we have two cars with different speeds. To find out when they meet, we use the distance formula for each car. We calculate the distance each car travels based on its speed and the time taken. This is formulated as:

• First car's distance: \( D_1 = \text{speed}_1 \times t \)
• Second car's distance: \( D_2 = \text{speed}_2 \times t \)

Understanding this concept is important because it lays the foundation for solving problems involving objects moving towards or away from each other. The key takeaway here is that the distance covered by each car is a product of its speed and the time traveled, allowing us to set equations and solve them.

Speed and Velocity

Speed and velocity are often confused, but they are distinct concepts. Speed is a scalar quantity indicating how fast an object moves, regardless of direction. Velocity, on the other hand, is a vector quantity that includes both speed and direction. In the exercise, both cars have specific speeds:

• First car: 40 miles per hour
• Second car: 50 miles per hour

These numbers tell us how fast each car is moving without indicating the direction, so they are related to speed. Understanding these concepts helps us when comparing the two cars' motions to determine how one may overtake the other. While velocity would be used to understand the exact motion in a given direction, focusing on speed here eases solving the problem as we are concerned with their rates of motion.

Equations of Motion

The primary reason the second car overtakes the first is due to the equations of motion. This involves setting up equations for each car’s motion and finding the intersection point in their journeys. We consider the speeds and initial conditions:

• The second car starts 5 miles behind.
• We equate the distances traveled by both cars, adjusting for the 5-mile difference: \( D_2 = D_1 + 5 \).

Substituting our earlier expressions for distance into this equation, we get:\[50t = 40t + 5\]By solving this equation, we find \( t \), the time it takes for the second car to catch up. This intersection point is critical as it shows when and where the overtaking happens. The solution to the equation tells us the second car catches up in 30 minutes, which reveals both the impact of speed on motion and the application of equations to predict future positions.