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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.8, Problem 3E

(a)

To determine

**To find:** The velocity at time *t*.

Expert Solution

The velocity at time is

**Given:**

The given equation is as below.

**Calculation:**

Calculate the velocity at time

Differentiate the equation (1) with respect to time.

Therefore, the velocity at time

(b)

To determine

**To find:** The velocity after 1 second.

Expert Solution

The velocity after 1 second is

Calculate the velocity after 1 second.

Substitute 1 for

Therefore, the velocity after 1 second is

(c)

To determine

**To find:** The time when particle at rest.

Expert Solution

The particle never is at rest when

Calculate the time when particle will be at rest.

The velocity will be zero, when the particle is at rest.

Substitute 0 for

Here,

(d)

To determine

**To find:** The particle moving in the positive direction.

Expert Solution

The velocity of particle always moves in positive direction when

Calculate the time at which the particle will be moving in the positive direction.

The particle will move in positive direction when

(e)

To determine

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

Expert Solution

The total distance travelled during first 6 second is

Calculate the total distance traveled during first 6 seconds.

Here, the velocity changes from 1, 3 and 5 which appears in the interval of [0,6].

Substitute 1 and 0 for

The total distance travelled is as below.

Therefore, the total distance travelled during first 6 seconds is

(f)

To determine

**To find:** The diagram to illustrate the motion of the particle.

Expert Solution

The diagram to illustrate the motion of particle is shown in the figure (1).

Calculate the distance *s* using the expression.

Substitute 0 for

Calculate the value of *t* and *s* as shown in the table (1).

t | s |

0 | 0 |

1 | 1 |

3 | -1 |

5 | 1 |

6 | 0 |

Show the diagram to illustrate the motion of the particle as shown below in figure (1).

(g)

To determine

**To find:** The acceleration at time *t* and after 1 second.

Expert Solution

The acceleration at time is

Calculate the acceleration at any time *t.*

Differentiate the equation (2) with respect to *t.*

Therefore, the acceleration at any time is

Calculate the acceleration after 1 second.

Substitute 1 for

Therefore, the acceleration after 1 second is

(h)

To determine

**To sketch:** The graph the position, velocity, and acceleration function for

Expert Solution

The position, velocity, and acceleration functions are plotted for time limits

Calculate the position with respect to time using the formula.

Substitute 0 for

Similarly, calculate the remaining values.

Calculate the value of

0.00 | 0.00 |

0.50 | 0.71 |

1.00 | 1.00 |

1.50 | 0.71 |

2.00 | 0.00 |

2.50 | -0.71 |

3.00 | -1.00 |

3.50 | -0.71 |

4.00 | 0.00 |

4.50 | 0.71 |

5.00 | 1.00 |

5.50 | 0.71 |

6.00 | 0.00 |

Calculate the velocity using the expression.

Substitute 0 for

Similarly, calculate the remaining values.

Calculate the value of

0.00 | 1.57 |

0.50 | 1.11 |

1.00 | 0.00 |

1.50 | -1.11 |

2.00 | -1.57 |

2.50 | -1.11 |

3.00 | 0.00 |

3.50 | 1.11 |

4.00 | 1.57 |

4.50 | 1.11 |

5.00 | 0.00 |

5.50 | -1.11 |

6.00 | -1.57 |

Calculate the acceleration using the formula.

Substitute 0 for

Similarly, calculate the remaining values.

Calculate the value of

0.00 | 0.00 |

0.50 | -1.74 |

1.00 | -2.47 |

1.50 | -1.74 |

2.00 | 0.00 |

2.50 | 1.74 |

3.00 | 2.47 |

3.50 | 1.74 |

4.00 | 0.00 |

4.50 | -1.74 |

5.00 | -2.47 |

5.50 | -1.74 |

6.00 | 0.00 |

Draw the position as a function of time curve as shown in the Figure (2).

Draw the speed as a function of time curve as shown in the figure (3).

Draw the acceleration as a function of time curve as shown in the figure (4).

(i)

To determine

**To find:** The time when the particle is speeding up and slowing down.

Expert Solution

The acceleration is negative when the value of time is

Calculate the time when particle speeding up and slowing down.

Substitute 0 for

Here, variable

Therefore, the acceleration is negative when the value of time is