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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.8, Problem 4E

(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

Therefore, the particle at rest when

(d)

To determine

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

Expert Solution

The particle will be moving in the positive direction within the time limits

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

If the particle moves in positive direction, the velocity at any time *t* will be greater than zero.

Therefore, the particle will be moving in the positive direction within the time limits

(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.

Substitute 6 for

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 is shown in the figure (1).

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 time *t.*

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

Therefore, the acceleration at 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 using the formula.

Substitute 0 for

Similarly, calculate the remaining values.

Tabulate the value of

0 | 0 |

0.5 | 0.15163 |

1 | 0.36788 |

1.5 | 0.50204 |

2 | 0.54134 |

2.5 | 0.51303 |

3 | 0.44808 |

3.5 | 0.36992 |

4 | 0.29305 |

4.5 | 0.22496 |

5 | 0.16845 |

5.5 | 0.12362 |

6 | 0.08924 |

Calculate the velocity using the expression.

Substitute 0 for

Similarly, calculate the remaining values.

Tabulate the value of

0 | 0 |

0.5 | 0.4549 |

1 | 0.36788 |

1.5 | 0.16735 |

2 | 0 |

2.5 | -0.1026 |

3 | -0.1494 |

3.5 | -0.1585 |

4 | -0.1465 |

4.5 | -0.125 |

5 | -0.1011 |

5.5 | -0.0787 |

6 | -0.0595 |

Calculate the acceleration using the formula.

Substitute 0 for

Similarly, calculate the remaining values.

Tabulate the value of

0 | 2 |

0.5 | 0.15163 |

1 | -0.3679 |

1.5 | -0.3905 |

2 | -0.2707 |

2.5 | -0.1436 |

3 | -0.0498 |

3.5 | 0.00755 |

4 | 0.03663 |

4.5 | 0.04721 |

5 | 0.04717 |

5.5 | 0.04189 |

6 | 0.0347 |

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 positive when the value of time

Calculate the time when particle is speeding up and slowing down.

Substitute 0 for

Substitute

Substitute

Therefore, the acceleration is positive when the value of time