Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 82b

Chapter 2, Problem 82b

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.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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A rocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt falls off the side of the rocket. The bolt hits the ground 6.0 s later. What was the rocket's acceleration?

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Textbook Question

A rubber ball is shot straight up from the ground with speed v₀. Simultaneously, a second rubber ball at height h directly above the first ball is dropped from rest. At what height above the ground do the balls collide? Your answer will be an algebraic expression in terms of h, v₀, and g.

Textbook Question

A sprinter can accelerate with constant acceleration for 4.0 s before reaching top speed. He can run the 100 meter dash in 10.0 s. What is his speed as he crosses the finish line?

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Textbook Question

A rubber ball is shot straight up from the ground with speed v₀. Simultaneously, a second rubber ball at height h directly above the first ball is dropped from rest. What is the maximum value of h for which a collision occurs before the first ball falls back to the ground?

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Textbook Question

A good model for the acceleration of a car trying to reach top speed in the least amount of time is a_𝓍 = a_₀ ─ kv_𝓍, where a_₀ is the initial acceleration and k is a constant. Find an expression for the car's velocity as a function of time.

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Textbook Question

A good model for the acceleration of a car trying to reach top speed in the least amount of time is a_x = a₀ ─ kv_x, where a₀ is the initial acceleration and k is a constant. Find an expression for k in terms of a₀ and the car's top speed v_max.

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