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Chapter 2: Problem 16

One athlete in a race running on a long, straight track with a constant speed \(v_{1}\) is a distance \(d\) behind a second athlete running with a constant speed \(v_{2}\). (a) Under what circumstances is the first athlete able to overtake the second athlete? (b) Find the time \(t\) it takes the first athlete to overtake the second athlete, in terms of \(d, v_{1}\), and \(v_{2}\). (c) At what minimum distance \(d_{2}\) from the leading athlete must the finish line be located so that the trailing athlete can at least tie for first place? Express \(d_{2}\) in terms of \(d, v_{1}\), and \(v_{2}\) by using the result of part (b).

Short Answer

Expert verified
The trailing athlete can overtake the leading athlete if their speed \(v_{1}\) is greater than \(v_{2}\). The time it takes for them to do so is \(d/(v_{1} - v_{2})\). The minimum distance the finish line must be from the leading athlete for the first athlete to tie is \(d * v_{1}/(v_{1} - v_{2})\). These solutions are all under the assumption that \(v_{1} > v_{2}\).

Step by step solution

01

Relative speed

Relative speed is the speed of one object as observed from another. In this case, the first athlete needs to run faster than the second athlete to overtake him. Therefore, for overtaking to happen, the condition is \(v_{1} > v_{2}\).
02

Time calculation

The time \(t\) for the first athlete to overtake the second can be calculated by using the concept of relative speed. The trailing athlete will cover the distance \(d\) at a relative speed of \(v_{1} - v_{2}\). Therefore, we can use the formula 'time = distance/speed' to calculate the time. Substituting the given values, we get \(t = d/(v_{1} - v_{2})\). This assumes \(v_{1} > v_{2}\).
03

Condition for tying the race

For the trailing athlete to tie the race, they have to overtake the leading athlete right at the finish line. Running for time \(t\) at the speed \(v_{1}\), the trailing athlete will have covered the distance \(d_{2}\). We can use the formula 'distance = speed * time' to calculate this. Substituting the values, we get \(d_{2} = v_{1} * t\). Using the expression from Step 2, we can rewrite this as \(d_{2} = v_{1} * (d/(v_{1} - v_{2})) = d * v_{1}/(v_{1} - v_{2})\). This is the minimum distance the finish line needs to be away from the leading athlete for the first athlete to tie the race. This also assumes \(v_{1} > v_{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Speed
When we talk about constant speed, we refer to an athlete maintaining the same pace throughout their run. In this scenario, both athletes keep a steady speed. This is crucial because it simplifies calculations. Since there鈥檚 no acceleration or changing speeds to consider, we can directly use simple formulas to analyze their motion.

Constant speed is an essential assumption in many physics problems. It allows us to focus only on distance and time. Knowing this, if an athlete runs with a speed of 'v', it means they cover the same amount of distance in each unit of time.

The key takeaway is that constant speed means predictable and uniform motion 鈥� making the math easier! This is the foundation for calculating when one runner might overtake another.
Time Calculation
To calculate the time it takes for the first athlete to overtake the second, we need to consider their relative speeds. Relative speed deals with how fast one object is moving compared to another. This means if two athletes are running, the relative speed would be the difference between their speeds.

If the first athlete runs faster, the formula for the time it takes them to catch up is based on this relative speed:
  • First, ensure that the speed of the first athlete, \( v_1 \), is greater than that of the second athlete, \( v_2 \).
  • Then, use the formula for time: \( t = \frac{d}{v_1 - v_2} \). Here, 'd' is the initial distance between them.
This formula helps us express how long it will take for the trailing athlete to cover the distance separating them and eventually pull ahead.
Distance and Speed Formula
To understand the strategic location of the finish line, we rely on the distance and speed formula. This formula is useful because it ties together three main variables: distance, speed, and time. Simply put, if you know any two of these, you can find the third.

In our exercise, we want to determine the minimum distance \( d_2 \) the finish line should be from the leading athlete. This ensures the trailing athlete has enough track to at least tie.

We start with the following:
  • The distance the trailing athlete covers to reach the leader is \( t \times v_1 \).
  • Using the previously found time expression, this becomes \( d_2 = v_1 \times \left(\frac{d}{v_1 - v_2}\right) = \frac{d \cdot v_1}{v_1 - v_2} \).
Through this neat manipulation and understanding of constants, we can strategically set up the finish line, making sure both athletes have a fair shot at winning or tying.

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Most popular questions from this chapter

S An athlete swims the length \(L\) of a pool in a time \(t_{1}\) and makes the return trip to the starting position in a time \(t_{2}\). If she is swimming initially in the positive \(x\)-direction, determine her average velocities symbolically in (a) the first half of the swim, (b) the second half of the swim, and (c) the round trip. (d) What is her average speed for the round trip?

A speedboat increases its speed uniformly from \(v_{i}=\) \(20.0 \mathrm{~m} / \mathrm{s}\) to \(v_{f}=30.0 \mathrm{~m} / \mathrm{s}\) in a distance of \(2.00 \times 10^{2} \mathrm{~m}\). (a) Draw a coordinate system for this situation and label the relevant quantities, including vectors. (b) For the given information, what single equation is most appropriate for finding the acceleration? (c) Solve the equation selected in part (b) symbolically for the boat's acceleration in terms of \(v_{f}, v_{i}\), and \(\Delta x\). (d) Substitute given values, obtaining that acceleration. (e) Find the time it takes the boat to travel the given distance.

A jet plane has a takeoff speed of \(v_{10}=75 \mathrm{~m} / \mathrm{s}\) and can move along the runway at an average acceleration of \(1.3 \mathrm{~m} / \mathrm{s}^{2}\). If the length of the runway is \(2.5 \mathrm{~km}\), will the plane be able to use this runway safely? Defend your answer.

The current indoor world record time in the \(200-\mathrm{m}\) race is \(19.92 \mathrm{~s}\), held by Frank Fredericks of Namibia (1996), while the indoor record time in the one-mile race is \(228.5 \mathrm{~s}\), held by Hicham El Guerrouj of Morroco (1997). Find the mean speed in meters per second corresponding to these record times for (a) the \(200-\mathrm{m}\) event and (b) the one-mile event.

Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration less than \(800 \mathrm{~m} / \mathrm{s}^{2}\) lasting for any length of time will not cause injury, whereas an acceleration greater than \(1000 \mathrm{~m} / \mathrm{s}^{2}\) lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is \(0.40 \mathrm{~m}\) above the floor. If the floor is hardwood, the child's head is brought to rest in approximately \(2.0 \mathrm{~mm}\). If the floor is carpeted, this stopping distance is increased to about \(1.0 \mathrm{~cm}\). Calculate the magnitude and duration of the deceleration in both cases, to determine the risk of injury. Assume the child remains horizontal during the fall to the floor. Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate.
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