Question

Problem 6.33 An Object with mass 90 kg moved in outer space, when it was at location <15,-31,-8> its speed was 6.0 m/s. A single constant force 180, 340. -140> N acted on the object while the object moved from location 15,-31,-82 m to location <18, -26, -14 m. Then a different single constant force <150, 240, 150 N acted on the object while the object moved from location <18, -26, -14 m to location <24, -35, -9 m. What is the speed of the object at this final location? final speed m/s the tolerance is +/-596

Answer 1

For the first Force :

$\underset{F_{1}}{\rightarrow}$ = 180 i + 340 j - 140 k

$\underset{r_{i}}{\rightarrow}$ = initial position = 15 i -31 j - 8 k

$\underset{r_{f}}{\rightarrow}$ = final position = 18 i - 26 j - 14 k

$\underset{\Delta r}{\rightarrow}$ = displacement = $\underset{r_{f}}{\rightarrow}$ - $\underset{r_{i}}{\rightarrow}$ = (18 i - 26 j - 14 k) - (15 i -31 j - 8 k) = 3 i + 5 j - 6 k

v_i = initial speed at initial position = 6 m/s

v_f = final speed at final position = ?

Using work-change in kinetic energy theorem

$\underset{F_{1}}{\rightarrow}$.$\underset{\Delta r}{\rightarrow}$ = (0.5) m (v²_f - v²_i)

(180 i + 340 j - 140 k).(3 i + 5 j - 6 k) = (0.5) (90) (v²_f - (6)²)

3080 = (0.5) (90) (v²_f - (6)²)

v_f = 10.22 m/s

For the second Force :

$\underset{F_{2}}{\rightarrow}$ = 150 i + 240 j + 150 k

$\underset{r_{i}}{\rightarrow}$ = initial position = 18 i - 26 j - 14 k

$\underset{r_{f}}{\rightarrow}$ = final position = 24 i - 35 j - 9 k

$\underset{\Delta r}{\rightarrow}$ = displacement = $\underset{r_{f}}{\rightarrow}$ - $\underset{r_{i}}{\rightarrow}$ = (24 i - 35 j - 9 k) - (18 i - 26 j - 14 k) = 6 i - 9 j + 5 k

v_i = initial speed at initial position = 10.22 m/s

v_f = final speed at final position = ?

Using work-change in kinetic energy theorem

$\underset{F_{1}}{\rightarrow}$.$\underset{\Delta r}{\rightarrow}$ = (0.5) m (v²_f - v²_i)

(150 i + 240 j + 150 k).(6 i - 9 j + 5 k) = (0.5) (90) (v²_f - (10.22)²)

- 510 = (0.5) (90) (v²_f - (10.22)²)

v_f = 9.65 m/s

Final speed = 9.65 m/s