Careful measurements have been made of Olympic sprinters in the 100 meter dash. A quite realistic model is that the sprinter's velocity is given by v_𝓍 = a ( 1 - e⁻ᵇᵗ ) where t is in s, v_𝓍 is in m/s, and the constants a and b are characteristic of the sprinter. Sprinter Carl Lewis's run at the 1987 World Championships is modeled with a = 11.81 m/s and b = 0.6887 s⁻¹. Find an expression for the distance traveled at time t.

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Start by recalling the relationship between velocity and displacement. Velocity is the derivative of displacement with respect to time: \( v_x = \frac{dx}{dt} \). To find the distance traveled \( x(t) \), we need to integrate the velocity function with respect to time.

Substitute the given velocity function \( v_x = a (1 - e^{-bt}) \) into the integral: \( x(t) = \int v_x \, dt = \int a (1 - e^{-bt}) \, dt \).

Break the integral into two parts: \( x(t) = \int a \, dt - \int a e^{-bt} \, dt \). The first term is straightforward, and the second term requires the use of integration techniques for exponential functions.

Evaluate the first term: \( \int a \, dt = at \). For the second term, use the formula for the integral of an exponential function: \( \int e^{-bt} \, dt = -\frac{1}{b} e^{-bt} \). Thus, \( \int a e^{-bt} \, dt = -\frac{a}{b} e^{-bt} \).

Combine the results to get the final expression for \( x(t) \): \( x(t) = at + \frac{a}{b} e^{-bt} + C \), where \( C \) is the constant of integration. Since the sprinter starts at \( x = 0 \) when \( t = 0 \), substitute these values to solve for \( C \). This gives \( C = -\frac{a}{b} \). The final expression for the distance traveled is \( x(t) = at - \frac{a}{b} (1 - e^{-bt}) \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In the context of the sprinter's velocity model, the term e⁻ᵇᵗ represents a decay factor that influences how quickly the sprinter reaches their maximum velocity. Understanding exponential growth and decay is crucial for analyzing how the sprinter's speed changes over time.

Integration

Integration is a fundamental concept in calculus that allows us to find the area under a curve, which in this case represents the distance traveled over time. To find the distance traveled by the sprinter, we need to integrate the velocity function vₓ with respect to time t. This process will yield a function that describes the total distance covered as a function of time.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. Although the sprinter's motion is modeled with a velocity function that changes over time, the principles of kinematics still apply. By understanding how velocity relates to distance and time, we can derive the expression for distance traveled by integrating the velocity function.