Q. 27P At t = 532.0s after midnight, a ... [FREE SOLUTION]

91影视

Chapter 3: Q. 27P (page 124)

At t = 532.0s after midnight, a spacecraft of mass 1400 kg is located at position (3x10⁵, 7x10⁵,-4x10⁵ ) m, and at that time an asteroid whose mass is 7 x 10 ¹⁵ kg is located at position (9 x 10 ^ 5, - 3 x 10 ^ 5, - 12 x 10 ^ 5) m. There are no other objects nearby. (a) Calculate the (vector) force acting on the spacecraft. (b) At t=532.0: the spacecraft's momentum was ${\overset{鈫�}{p}}_{i}$ and at the later time t = 538.0s its momentum was ${\overset{鈫�}{p}}_{f}$ .Calculate the (vector) change of momentum ${\overset{鈫�}{p}}_{f} - {\overset{鈫�}{p}}_{i}$ .

Short Answer

Expert verified

The value of vector force $F = <7.26 脳 10^{3}, 0, 0>$
Change in vector momentum ${\overset{鈫�}{p}}_{f} - {\overset{鈫�}{p}}_{i} = <40 脳 10^{3}, 0, 0> Kg 路 m / s$

Step by step solution

Identification of given data

$t = 532.0 s$
Mass of spacecraft $m = 1400 k g$
Positions of spacecraft $(3 脳 10^{5}, 7 脳 10^{5}, - 4 脳 10^{5})$
Mass of asteroid ${\overset{鈫�}{p}}_{f} - {\overset{鈫�}{p}}_{i} = 40 脳 10^{3} N 路 m$

Calculation of the vector force

(a) According to the Newton鈥檚 law of gravitation

$F = \frac{G m_{1} m_{2}}{r^{2}}$

Where

G 鈥� Gravitational constant

$m_{1}$ - Mass of 1 object

$m_{2} -$ Mass of 2 objects

$r -$ Distance between two object

$F = \frac{6.67 脳 10^{- 11} 脳 1400 脳 7 脳 10^{15}}{{(3 脳 10^{5})}^{2}} F = 7.26 脳 10^{3} N F = 鉄� 7.26 脳 10^{3}, 0, 0 鉄� N$

Calculation of change in vector momentum

(b) The spacecraft momentum

When t=532 s

${\overset{鈫�}{p}}_{i} = F 路 t {\overset{鈫�}{p}}_{i} = 7.26 脳 10^{3} 脳 532 {\overset{鈫�}{p}}_{i} = 鉄� 3.86 脳 10^{6}, 0, 0 鉄� K g 鈰� m / s$

When t=538 s

${\overset{鈫�}{p}}_{f} = F 路 t {\overset{鈫�}{p}}_{f} = 7.26 脳 10^{3} 脳 538 {\overset{鈫�}{p}}_{f} = 鉄� 3.90 脳 10^{6}, 0, 0 鉄� K g 鈰� m / s$

Then the value of role="math" localid="1658079511166" ${\overset{鈫�}{p}}_{f} - {\overset{鈫�}{p}}_{i}$

${\overset{鈫�}{p}}_{f} - {\overset{鈫�}{p}}_{i} = 3.86 脳 10^{6} - 3.90 脳 10^{6} {\overset{鈫�}{p}}_{f} - {\overset{鈫�}{p}}_{i} = 鉄� 40 脳 10^{3}, 0, 0 鉄� K g 鈰� m / s$