Question

Suppose that in a race, horse \(A\) and horse \(B\) begin at the same point and finish in a dead heat. Prove that their speeds were identical at some instant of the race.

Short answer

The Intermediate Value Theorem implies their speeds were identical at some instant.

Step-by-step solution

Understand the Problem

Both horse A and horse B start at the same point in the race, and they both finish at the same time. We need to show that at some moment during the race, their speeds were equal.

Apply Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that for any continuous function that takes on values at two endpoints, it also takes on any value between those two endpoints. In our case, this involves finding a moment where the difference in speeds is zero.

Define the Speed Functions

Let the speed of horse A at any time \(t\) be \(v_A(t)\) and the speed of horse B be \(v_B(t)\). The functions \(v_A(t)\) and \(v_B(t)\) are continuous because speed is a continuous function of time.

Define the Speed Difference Function

Consider the function \(f(t) = v_A(t) - v_B(t)\). This function represents the difference in speeds between horse A and horse B at time \(t\).

Evaluate at Endpoints

At the start of the race, \(f(0) = v_A(0) - v_B(0)\). At the end of the race, since they finish together, \(\int v_A(t) \, dt = \int v_B(t) \, dt\). Thus, the net difference in total distance is zero, suggesting an intermediate moment where the speed difference was zero.

Use the Mean Value Theorem for Integrals

Since the integrals over their speeds are equal, by the Mean Value Theorem for Integrals, there must exist a point \(c\) in the interval where \(v_A(c) = v_B(c)\). This is equivalent to saying \(f(c) = 0\), i.e., their speeds are equal.

Conclusion

Based on the Mean Value Theorem, there must be at least one instant where the speed of horse A equals the speed of horse B.

Key concepts

Speed Functions

In mathematical terms, a speed function describes how an object's speed varies over time. For horse races, if we designate the speed of horse A at any time \(t\) as \(v_A(t)\) and the speed of horse B as \(v_B(t)\), these functions provide a continuous record of their respective speeds throughout the race.
The continuity of these functions is essential for applying theorems that predict certain behaviors in their speeds. To put it simply, a continuous speed function means the horse doesn't suddenly vanish and reappear in another place. Instead, it shows a smooth progression from one point to another over time.

• Speed functions help track changes in speed over time.
• Continuity is necessary for mathematical predictions and proofs.

Because horse races involve continuous movements, each horse's speed function remains uninterrupted from start to finish.

Mean Value Theorem

The Mean Value Theorem (MVT) is a crucial concept in calculus that connects average rates of change with instantaneous rates of change. When discussing horse races, the MVT helps us understand why horse A and horse B must have had identical speeds at some instant during the race.
The theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point within that interval where the derivative of the function equals the average rate of change over the interval.
For horse races:

• The "average rate of change" can be the average speed.
• The "derivative" represents the actual speed at any given point.

Applying the MVT indicates that if two horses start and finish the race at the same time, their speed graphs must intersect at least once - highlighting a moment when their speeds are equal.

Continuous Function

The notion of a continuous function is foundational in mathematics, especially when applying the Intermediate Value Theorem (IVT) or the Mean Value Theorem. A function is continuous if it is uninterrupted or has no gaps or jumps in its graph.
In the context of horse races, each horse's speed function, \(v_A(t)\) or \(v_B(t)\), is continuous because the speed changes smoothly as the race progresses. This characteristic of being continuous allows us to apply theorems concerning the speed difference functions of horse A and B.

• Continuity ensures smooth transitions without abrupt changes.
• Essential for predicting and proving certain moments of equal speeds.

Without continuity, the mathematical predictions about the race could fail, as jumps in speeds would invalidate many calculus-based analyses.

Difference in Speeds

Analyzing the difference in speeds between two horses requires understanding a new function: \(f(t) = v_A(t) - v_B(t)\). This function tells us how horse A's speed compares to horse B's at any given moment in time.
Looking at the endpoints of this function during the race:

• At the start: It might not be zero as horses can start at different speeds.
• At the end: It sums to zero since the horses finish simultaneously.

The idea here is to find an instant \(c\) during which \(f(c) = 0\), meaning both horses had the same speed. The Intermediate Value Theorem helps us deduce that because the net difference in speed over time is zero, there must be a point where both speeds align.
This "point \(c\)" is a pivotal moment proving that at some instant in time, horse A and horse B were not just neck and neck but also pace and pace.