   Chapter 2.7, Problem 2E

Chapter
Section
Textbook Problem

# A particle moves according to a law of motion s   =   f ( t ) ,   t ≥   0 , where t is measured in seconds and s in feet.(a) Find the velocity at time t.(b) What is the velocity after 1 second?(c) When is the particle at rest?(d) When is the particle moving in the positive direction?(e) Find the total distance traveled during the first 6 seconds.(f) Draw a diagram like Figure 2 to illustrate the motion of the particle.(g) Find the acceleration at time t and after 1 second.(h) Graph the position, velocity, and acceleration functions for 0 ≤ t ≤ 6 .(i) When is the particle speeding up? When is it slowing down? f ( t ) = 0.01 t 4 + 0.04 t 3

To determine

(a)

To find: The velocity at time t

Explanation

Concept:

Velocity is rate of change of position with respect to time t.

Formula:

Power rule,

ddxxn=n*xn-1

Given:

ft=0.01t4-0.04t3

Calculation:

Position at time t is,ft=0.01t4-0.04t3

The velocity at time t is the rate of change of position with respect to time t, that is derivative of f (t) with respect to  t,

By differentiating f(t) we get,

f't=ddt(0.01t4-0.04t3)

By difference rule,

ddxfx-gx=ddxfx-ddx(gx)

f't=ddt(0

To determine

(b)

To find: Velocity after 1 second

To determine

(c)

To find: Time (t) when the particle is at rest

To determine

(d)

To find: Time (t) when the particle is moving in positive direction

To determine

(e)

To find: The total distance traveled during the first 6 seconds

To determine

(f)

To find: Diagram to illustrate the motion of the particle

To determine

(g)

To find: Acceleration at time t and after 1 second

To determine

(h)

To find: Graph the position, velocity and acceleration functions for 0 ≤ t ≤ 6

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