Concept explainers
Students are testing their new drone to see if it can safely deliver packages to different departments on campus. Position data can be approximated using the expressions x(t) = −0.0000225t4 + 0.003t3 + 0.01t2 and
(a)
Plot the path of the drone and find the duration (t) of the flight.
Answer to Problem 11.182RP
The duration (t) of the flight is
Explanation of Solution
Given information:
The x coordinate is defined by the relation as
The y coordinate is defined by the relation as
Calculation:
The x coordinate is defined by the relation:
The y coordinate is defined by the relation:
Calculate the duration (t) of the flight:
Equate equation (2) to zero.
General solution for
Calculate the x coordinated as time (t) 0 sec.
Substitute 0 for t in Equation (1).
Similarly calculate the x coordinate for time interval of
Tabulate the calculated values of x coordinate for time interval
Time (t)(sec) | x(m) |
0 | 0.00 |
5 | 0.61 |
10 | 3.78 |
15 | 11.24 |
20 | 24.40 |
25 | 44.34 |
30 | 71.78 |
35 | 107.11 |
40 | 150.40 |
45 | 201.36 |
50 | 259.38 |
55 | 323.49 |
60 | 392.40 |
65 | 464.49 |
70 | 537.78 |
75 | 609.96 |
80 | 678.40 |
Plot the graph for time (t) and x coordinate as in Figure (1).
Calculate the y coordinated as time (t) 0 sec.
Substitute 0 for t in equation (1).
Similarly calculate the y coordinate for time interval of
Tabulate the calculated values of y coordinate for time interval
Time (t)(sec) | y(m) |
0 | 0.00 |
5 | 22.84 |
10 | 87.87 |
15 | 185.19 |
20 | 300.00 |
25 | 414.81 |
30 | 512.13 |
35 | 577.16 |
40 | 600.00 |
45 | 577.16 |
50 | 512.13 |
55 | 414.81 |
60 | 300.00 |
65 | 185.19 |
70 | 87.87 |
75 | 22.84 |
80 | 0.00 |
Plot the graph for time (t) and y coordinate as in Figure (2).
Tabulate the x and y coordinates value as in Table (3):
x(m) | y(m) |
0.00 | 0.00 |
0.61 | 22.84 |
3.78 | 87.87 |
11.24 | 185.19 |
24.40 | 300.00 |
44.34 | 414.81 |
71.78 | 512.13 |
107.11 | 577.16 |
150.40 | 600.00 |
201.36 | 577.16 |
259.38 | 512.13 |
323.49 | 414.81 |
392.40 | 300.00 |
464.49 | 185.19 |
537.78 | 87.87 |
609.96 | 22.84 |
678.40 | 0.00 |
Plot the graph for coordinate x and y as in Figure (3).
Therefore, the duration (t) of the flight is
(b)
The maximum speed
Answer to Problem 11.182RP
The maximum speed
Explanation of Solution
Given information:
The x coordinate is defined by the relation as
The y coordinate is defined by the relation as
Calculation:
Differentiate equation (1) with respective to time (t).
Since, the rate of change of any coordinate with respect to time is equal to the velocity.
Differentiate equation (3) with respective to time (t).
Since, the rate of change of velocity with respect to time is equal to the acceleration.
Calculate the time (t) at which the velocity is maximum:
Equate the equation (4) to zero,
Solve the above quadratic equation for the roots (t),
The roots are -1.093 sec and 67.76 sec. Reject the negative root.
Calculate the maximum speed
Substitute 67.76 sec for t in equation (3).
Therefore, the maximum speed
(c)
The maximum altitude
Answer to Problem 11.182RP
The maximum altitude
Explanation of Solution
Given information:
The x coordinate is defined by the relation as
The y coordinate is defined by the relation as
Calculation:
Calculate the maximum altitude
Refer Figure 2, the maximum altitude 600m at time 40 sec.
Substitute 40 sec in equation (2).
Calculate the horizontal
Substitute 80 sec for t in equation (1).
Therefore, the maximum altitude
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