Question

In the 2016 Olympics, Usain Bolt of Jamaica won the gold medal in the 100 -meter race with a time of 9.81 seconds. In the 1896 Olympics, Thomas Burke of the United States won the gold medal in the 100-meter race in 12.0 seconds. If they ran in the same race, repeating their respective times, by how many meters would Bolt beat Burke?

Short answer

Bolt would beat Burke by approximately 18.24 meters.

Step-by-step solution

- Determine Bolt's Speed

Calculate Usain Bolt's speed by dividing the distance he ran by his time. Bolt's speed is given by the formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] where the distance is 100 meters and the time is 9.81 seconds. Thus, Bolt's speed is \[ \text{Speed}_{\text{Bolt}} = \frac{100}{9.81} \approx 10.19 \text{ meters/second} \]

- Determine Burke's Speed

Calculate Thomas Burke's speed by using the same formula: \[ \text{Speed}_{\text{Burke}} = \frac{100}{12.0} \approx 8.33 \text{ meters/second} \]

- Calculate Time Difference

Find the difference in their times by subtracting Bolt's time from Burke's time: \[ \text{Time}_\text{difference} = 12.0 - 9.81 = 2.19 \text{ seconds} \]

- Calculate Distance Covered by Burke in Time Difference

Find how many meters Burke would run in the time difference. This is done by multiplying Burke's speed by the time difference: \[ \text{Distance}_\text{Burke} = \text{Speed}_\text{Burke} \times \text{Time}_\text{difference} \] which gives \[ \text{Distance}_\text{Burke} = 8.33 \times 2.19 \approx 18.24 \text{ meters} \]

- Calculate Meters Bolt Would Win By

Since we are asked by how many meters Bolt would win, the answer is the distance calculated in the previous step: \[ \text{Bolt wins by } \approx 18.24 \text{ meters} \]

Key concepts

distance-time relationship

The distance-time relationship in speed calculations is fundamental. Speed is a measure of how fast an object moves and can be determined if you know the distance covered and the time taken. The basic formula for speed is: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\). For instance, if Usain Bolt ran 100 meters in 9.81 seconds, we use the formula to find his speed. This relationship highlights how faster speeds reduce travel time for the same distance.

unit conversion

In speed calculation problems, it's essential to ensure that units are consistent. Typically, distances might be given in meters and times in seconds, so the speed will be in meters per second (m/s). However, if distances were given in different units like kilometers or miles, we would need to convert them to meters first. Consistent units allow us to apply formulas accurately and avoid mistakes in calculations. Remember, always check your units before finalizing your solution.

mathematical comparison

Making mathematical comparisons can help understand how one value relates to another. In our exercise, we compared the speeds of two athletes and their race times. We calculated their speeds and then found the time difference by subtracting one time from the other: \(\text{Time}_\text{difference} = 12.0 \text{ s} - 9.81 \text{ s} = 2.19 \text{ s}\). Finally, using Burke’s speed and the time difference, we determined how many more meters Bolt would have run: \(\text{Distance}_\text{Burke} = 8.33 \text{ m/s} \times 2.19 \text{ s} \approx 18.24 \text{ meters}\).

linear equations

Linear equations are a straightforward method used to solve problems involving constant speed, distance, and time. In this exercise, you saw how we used linear equations to determine each athlete’s speed: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\). Then, by finding the time difference and multiplying by Burke's speed, we employed another linear equation to calculate the additional distance Bolt would cover: \(\text{Distance}_\text{Burke} = \text{Speed}_\text{Burke} \times \text{Time}_\text{difference}\). Linear equations are helpful tools in breaking down these problems into manageable steps.