Table of contents

0. Math Review31m
- Math Review
  31m
1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

2. 1D Motion / Kinematics

Vectors, Scalars, & Displacement

Physics

2. 1D Motion / Kinematics

Vectors, Scalars, & Displacement

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5:09 minutes

Problem 82b

Textbook Question

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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Related Practice

Textbook Question

When a 1984 Alfa Romeo Spider sports car accelerates at the maximum possible rate, its motion during the first 20 s is extremely well modeled by the simple equation v_x² = (2P/m)t, where P = 3.6 ✕ 10⁴ watts is the car's power output, m = 1200 kg is its mass, and v_x is in m/s. That is, the square of the car's velocity increases linearly with time. Find an algebraic expression in terms of P, m, and t for the car's acceleration at time t.

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Textbook Question

A car's velocity as a function of time is given by $v_x(t) = α + βt^2$ , where $α = 3.00$ m/s and $β = 0.100$ m/s³. Draw $v_{x}$ - $t$ and $a_{x}$ - $t$ graphs for the car's motion between $t equals 0$ and $t equals 5.00$ s.

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Textbook Question

A turtle crawls along a straight line, which we will call the $x$ -axis with the positive direction to the right. The equation for the turtle's position as a function of time is $x(t) = 50.0$ cm + ( $2.00$ cm/s) $t$ − ( $0.0625$ cm/s²) $t^2$ . Sketch graphs of $x$ versus $t$ , $v_{x}$ versus $t$ , and $a_{x}$ versus $t$ , for the time interval $t equals 0$ to $t equals 40$ s.

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Textbook Question

A rubber ball bounces. We'd like to understand how the ball bounces. A rubber ball has been dropped and is bouncing off the floor. Draw a motion diagram of the ball during the brief time interval that it is in contact with the floor. Show 4 or 5 frames as the ball compresses, then another 4 or 5 frames as it expands. What is the direction of a during each of these parts of the motion?

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Multiple Choice

Starting from a pillar, you run a distance 140m east (the + x-direction), then turn around.

(a) How far west would you have to walk so that your total distance traveled is 300m?

(b) What is the magnitude and direction of your total displacement?