Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.5.51a

Chapter 4, Problem 4.5.51a

51. Frictionless cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 10 cos πt.

a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then?

Verified step by step guidance

To find the cart's maximum speed, we need to determine the derivative of the position function s(t) = 10 cos(πt) with respect to time t. This derivative will give us the velocity function v(t).

The velocity function v(t) is found by differentiating s(t) with respect to t: v(t) = ds/dt = -10π sin(πt).

The maximum speed occurs when the absolute value of the velocity is maximized. Since the sine function oscillates between -1 and 1, the maximum value of |v(t)| is 10π, which occurs when sin(πt) = ±1.

To find when the cart is moving at maximum speed, solve the equation sin(πt) = ±1. This occurs at t = 0.5, 1.5, 2.5, 3.5, etc., within the given time interval of 0 to 4 seconds.

To find the position and acceleration when the cart is moving at maximum speed, substitute t = 0.5 into the position function s(t) and the acceleration function a(t), where a(t) = dv/dt = -10π^2 cos(πt).

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Key Concepts

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

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the motion of the cart as it oscillates back and forth. The position function s = 10 cos πt indicates that the cart's motion is periodic and sinusoidal, typical of SHM. The amplitude of 10 cm represents the maximum displacement from the rest position, and the angular frequency π determines the speed of oscillation.

Velocity in SHM

The velocity of an object in simple harmonic motion is the derivative of its position function with respect to time. For s = 10 cos πt, the velocity v(t) is given by v(t) = -10π sin πt. The maximum speed occurs when the sine function equals ±1, resulting in a maximum velocity magnitude of 10π cm/s.

Acceleration in SHM

Acceleration in simple harmonic motion is the derivative of the velocity function or the second derivative of the position function. For s = 10 cos πt, the acceleration a(t) is a(t) = -10π² cos πt. The magnitude of acceleration is maximum when the cosine function equals ±1, which occurs at the maximum displacement points.

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Related Practice

Textbook Question

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

g(x) = −x² − 6x − 9,−4 ≤ x < ∞

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

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Textbook Question

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = √3cos x + sin x, 0 ≤ x ≤ 2π

Textbook Question

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = (x − 1)²(x + 2)

Textbook Question

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(t) = t³ − 3t², −∞ < t ≤ 3

Textbook Question

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = (x − 1)(x + 2)(x − 3)