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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.8, Problem 2E

(a)

To determine

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

Expert Solution

The velocity at time *t* is

**Given:**

The given equation is as below.

**Calculation:**

Calculate the velocity at time

Differentiate the equation (1) with respect to time by applying

Therefore, the velocity at time *t* is

(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 time when particle at rest is

Calculate the time when particle at rest.

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

Substitute 0 for

Therefore, the time when particle at rest is

(d)

To determine

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

Expert Solution

The particle always moves in positive direction when time is within the range

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 moving in the 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 particle moves in positive and negative direction, the total distance traveled should be calculated between the intervals of

Substitute 3 and 0 for

Substitute 6 and 3 for *t* between intervals of

The total distance travelled is as given below.

Therefore, the total distance travelled during first 6 second is

(f)

To determine

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

Expert Solution

Show the diagram to illustrate the motion of the particle as shown below in Figure 1.

Figure 1 shows the movement of particle at different times.

(g)

To determine

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

Expert Solution

The acceleration at time *t* is

Calculate the acceleration at time *t.*

Differentiate the equation (2) with respect to *t* by applying

On further simplification,

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

Expert Solution

Calculate the position of the particle with respect to time using the expression.

Substitute 0 for

Similarly, calculate the remaining values.

Calculate the value of

0 | 0 |

0.5 | 0.48649 |

1 | 0.9 |

1.5 | 1.2 |

2 | 1.38462 |

2.5 | 1.47541 |

3 | 1.5 |

3.5 | 1.48235 |

4 | 1.44 |

4.5 | 1.38462 |

5 | 1.32353 |

5.5 | 1.26115 |

6 | 1.2 |

Calculate the velocity using the formula.

Substitute 0 for

Similarly, calculate the remaining values.

Calculate the value of

0 | 1 |

0.5 | 0.92038 |

1 | 0.72 |

1.5 | 0.48 |

2 | 0.26627 |

2.5 | 0.10642 |

3 | 0 |

3.5 | -0.0648 |

4 | -0.1008 |

4.5 | -0.1183 |

5 | -0.1246 |

5.5 | -0.1241 |

6 | -0.12 |

Calculate the acceleration using the formula.

Substitute 0 for

Similarly, calculate the remaining values.

Calculate the value of

0 | 0.00 |

0.5 | -0.32 |

1 | -0.47 |

1.5 | -0.35 |

2 | -0.17 |

2.5 | -0.06 |

3 | -0.02 |

3.5 | -0.01 |

4 | 0.00 |

4.5 | 0.00 |

5 | 0.00 |

5.5 | 0.00 |

6 | 0.00 |

Draw the graph of the position, velocity and acceleration functions as shown in the Figure 2.

(i)

To determine

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

Expert Solution

the acceleration is zero when time *t* is greater than *t* is less than

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

Substitute 0 for

Substitute

Substitute

Therefore, the acceleration is zero when time *t* is greater than *t* is less than