   Chapter 3.1, Problem 50E

Chapter
Section
Textbook Problem

The equation of motion of a particle iss = t4 – 2t3 + t2 – t, where·s is in meters and t is in seconds.(a) Find the velocity and acceleration as functions of t.(b) Find the acceleration after 1 s.(c) Graph the position, velocity, and acceleration functions on the same screen.

(a)

To determine

To find: The velocity and acceleration as functions of t.

Explanation

Given:

The equation of motion of a particle is s=t42t3+t2t, where s is in meters and t is in seconds.

Derivative rules:

(1) Constant Function: ddx(c)=0

(2) Constant Multiple Rule: ddx[cf(x)]=cddxf(x)

(3) Power Rule: ddx(xn)=nxn1

(4) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))

(5) Difference Rule: ddx[f(x)g(x)]=ddx(f(x))ddx(g(x))

Recall:

If s is a displacement of a particle and the time t is in seconds, then the velocity of the particle is v=dsdt.

If v is a velocity of the particle and the time t is in seconds, then the acceleration of the particle is a=dvdt.

Calculation:

Obtain the velocity of the particle.

v=ddt(s) =ddt(t42t3+t2t)

Apply the sum rule (4) and the difference rule (5),

v=ddt(t4)ddt(2t3)+ddt(t2)ddt(t)

Apply the constant multiple rule (2) and the power rule (3),

v=ddt(t4)2ddt(t3)+ddt(t2)ddt(

(b)

To determine

To find: The acceleration of the particle after 1 second.

(c)

To determine

To sketch: The graph the position, velocity and the acceleration of the functions.

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