# The velocity of the particle at time t. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.4, Problem 70E

(a)

To determine

## To find: The velocity of the particle at time t.

Expert Solution

The velocity of the particle at time t is Awsin(wt+δ)_.

### Explanation of Solution

Given:

The equation of motion of a particle is s=Acos(wt+δ).

Derivative rule:

Constant Multiple Rule:

If c is a constant and f(x) is a differentiable function, then

ddx[cf(x)]=cddx[f(x)] (1)

Formula used: Chain Rule

If g is differentiable at x and f is differentiable at g(x), then the composite function F=fg defined by F(x)=f(g(x)) is differentiable at x and F is given by the product

F(x)=f(g(x))g(x) (2)

Recall:

If x(t) is the displacement of a particle and the time t is in seconds, then the velocity of the particle is v(t)=dxdt.

Calculation:

Obtain the velocity of the particle at time t.

v(t)=ddt(s(t))=ddt(Acos(wt+δ))

Let g(t)=wt+δ and f(u)=Acosu  where u=g(t).

Apply the chain rule as shown in equation (2),

v(t)=f(g(t))g(t) (3)

The derivative of f(g(t)) is computed as follows,

Simplify further,

f(g(t))=A(sinu)=Asinu

Substitute u=wt+δ in the above equation,

f(g(t))=Asin(wt+δ)

Thus, the derivative f(g(t))=Asin(wt+δ).

The derivative of g(t) is computed as follows,

g(t)=ddt(wt+δ)=ddt(wt)+ddt(δ)=w+0=w

Thus, the derivative g(t)=w.

Substitute Asin(wt+δ) for f(g(t)) and w for g(t) in equation (3),

v(t)=Asin(wt+δ)(w)=Awsin(wt+δ)

Therefore, the velocity of particle at time t is v(t)=Awsin(wt+δ)_.

(b)

To determine

### To find: The time such that the velocity is 0.

Expert Solution

The velocity is 0 when t=nπδw, n is an integer.

### Explanation of Solution

Calculation:

From part (a), the velocity of particle is v(t)=Awsin(wt+δ).

Since the velocity is zero, v(t)=0.

That is, Awsin(wt+δ)=0.

If A0 and w0, then sin(wt+δ)=0.

sin(wt+δ)=0sin1(sin(wt+δ))=sin1(0)wt+δ=sin1(0)

For n, sin1(0)=nπ.

wt+δ=nπwt=nπδt=nπδw

Therefore, the velocity is 0 when t=nπδw, n is an integer.

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