Analysis of Race Performance: Determining the Time Taken by A to Finish a 1200-Meter Race

In competitive running, understanding the performance metrics like speed, time, and distance can provide valuable insights into the athletes' abilities. This article delves into the analysis of a race result where Athlete A (A) finishes 30 meters or 10 seconds ahead of Athlete B (B) in a 1200-meter race. Let's delve into how we can calculate the time taken by Athlete A to complete the race, using the principles of speed, distance, and time.

Understanding the Given Information

We are given that Athlete A finishes the race 30 meters ahead of Athlete B, and it takes Athlete B an additional 10 seconds to cover those 30 meters. This means that while Athlete A runs a full 1200 meters, Athlete B only covers 1170 meters. To calculate the time taken by Athlete A to complete the race, we will use the principles of speed, distance, and time.

Step-by-Step Analysis

Step 1: Express the time taken by B to complete the race in terms of A's time.

Let's denote the time taken by Athlete A to complete the race as ( t_A ). Since Athlete B runs 1170 meters in ( t_B ), and we know that:

( t_B t_A 10 )

Step 2: Relate the distances and speeds of A and B.

Using the formula for speed:

( text{Speed} frac{text{Distance}}{text{Time}} )

For Athlete A:

( v_A frac{1200}{t_A} )

For Athlete B:

( v_B frac{1170}{t_B} )

Since ( t_B t_A 10 ), we can substitute this into the equation for Athlete B's speed:

( v_B frac{1170}{t_A 10} )

Step 3: Equate the speeds of A and B.

Since they are running at their respective speeds, we can set the two speed equations equal to each other:

( frac{1200}{t_A} frac{1170}{t_A 10} )

Step 4: Cross-multiply to solve for ( t_A ).

Cross-multiplying gives:

( 1200(t_A 10) 1170t_A )

Expanding this:

( 1200t_A 12000 1170t_A )

Step 5: Rearrange the equation to isolate ( t_A ).

Rearranging the equation gives:

( 1200t_A - 1170t_A -12000 )

( 30t_A 12000 )

( t_A frac{12000}{30} 400 text{ seconds} )

Conclusion

Athlete A took 400 seconds to complete the race of 1200 meters. This analysis shows that understanding the performance metrics in competitive running is crucial for evaluating and improving one's abilities.

To summarize, Athlete B's velocity was calculated to be (30/10 3) meters/second, and at this speed, it would take 400 seconds for Athlete B to cover 1200 meters. Since Athlete A finished 10 seconds earlier, Athlete A completed the race in 390 seconds.

The results also indicate that 1200 meters consists of 40 segments of 30 meters, which Athlete B covers in 10 seconds, meaning B's velocity is 3 meters/second. Given this, the total time for Athlete B to cover 1200 meters is 400 seconds, and thus, Athlete A finished in 390 seconds (400 - 10).