Answer :
To solve this problem, let's break it down step-by-step:
### Part (a):
1. Given Values:
- Mass of the boat, [tex]\( m = 100,000,000 \text{ kg} \)[/tex]
- Force exerted by the tugboat, [tex]\( F_{\text{tugboat}} = 900,000 \text{ Newtons} \)[/tex]
- Final speed, [tex]\( v = 0.2 \text{ m/s} \)[/tex]
- Time taken to reach this speed, [tex]\( t = 10 \times 60 = 600 \text{ seconds} \)[/tex]
2. Calculate the acceleration ([tex]\( a \)[/tex]):
[tex]\[
a = \frac{v}{t} = \frac{0.2 \text{ m/s}}{600 \text{ s}} = 0.000333 \text{ m/s}^2
\][/tex]
3. Calculate the inertia force ([tex]\( I \)[/tex]):
[tex]\[
I = m \times a = 100,000,000 \text{ kg} \times 0.000333 \text{ m/s}^2 = 33,333.33 \text{ Newtons}
\][/tex]
4. Calculate the drag force ([tex]\( F_{\text{drag}} \)[/tex]):
- The tugboat force has to overcome the inertia and the drag force from the water.
- Therefore, [tex]\( F_{\text{tugboat}} = I + F_{\text{drag}} \)[/tex]
[tex]\[
F_{\text{drag}} = F_{\text{tugboat}} - I = 900,000 \text{ N} - 33,333.33 \text{ N} = 866,666.67 \text{ Newtons}
\][/tex]
### Part (b):
Now, let's determine how long it would take if the tugboat exerts a different force:
1. New force exerted by the tugboat:
[tex]\[
F_{\text{tugboat (new)}} = 1,100,000 \text{ Newtons}
\][/tex]
2. Calculate the net force (after considering drag force):
[tex]\[
F_{\text{net new}} = F_{\text{tugboat (new)}} - F_{\text{drag}} = 1,100,000 \text{ N} - 866,666.67 \text{ N} = 233,333.33 \text{ Newtons}
\][/tex]
3. Calculate the new acceleration ([tex]\( a_{\text{new}} \)[/tex]):
[tex]\[
a_{\text{new}} = \frac{F_{\text{net new}}}{m} = \frac{233,333.33 \text{ N}}{100,000,000 \text{ kg}} = 0.002333 \text{ m/s}^2
\][/tex]
4. Calculate the time to reach the speed of [tex]\( 0.2 \text{ m/s} \)[/tex] with the new force:
[tex]\[
v = a_{\text{new}} \times t_{\text{new}}
\][/tex]
[tex]\[
t_{\text{new}} = \frac{v}{a_{\text{new}}} = \frac{0.2 \text{ m/s}}{0.002333 \text{ m/s}^2} \approx 85.71 \text{ seconds}
\][/tex]
Thus, it would take approximately 85.71 seconds for the boat to reach the same speed if the new tugboat force is applied.
### Part (a):
1. Given Values:
- Mass of the boat, [tex]\( m = 100,000,000 \text{ kg} \)[/tex]
- Force exerted by the tugboat, [tex]\( F_{\text{tugboat}} = 900,000 \text{ Newtons} \)[/tex]
- Final speed, [tex]\( v = 0.2 \text{ m/s} \)[/tex]
- Time taken to reach this speed, [tex]\( t = 10 \times 60 = 600 \text{ seconds} \)[/tex]
2. Calculate the acceleration ([tex]\( a \)[/tex]):
[tex]\[
a = \frac{v}{t} = \frac{0.2 \text{ m/s}}{600 \text{ s}} = 0.000333 \text{ m/s}^2
\][/tex]
3. Calculate the inertia force ([tex]\( I \)[/tex]):
[tex]\[
I = m \times a = 100,000,000 \text{ kg} \times 0.000333 \text{ m/s}^2 = 33,333.33 \text{ Newtons}
\][/tex]
4. Calculate the drag force ([tex]\( F_{\text{drag}} \)[/tex]):
- The tugboat force has to overcome the inertia and the drag force from the water.
- Therefore, [tex]\( F_{\text{tugboat}} = I + F_{\text{drag}} \)[/tex]
[tex]\[
F_{\text{drag}} = F_{\text{tugboat}} - I = 900,000 \text{ N} - 33,333.33 \text{ N} = 866,666.67 \text{ Newtons}
\][/tex]
### Part (b):
Now, let's determine how long it would take if the tugboat exerts a different force:
1. New force exerted by the tugboat:
[tex]\[
F_{\text{tugboat (new)}} = 1,100,000 \text{ Newtons}
\][/tex]
2. Calculate the net force (after considering drag force):
[tex]\[
F_{\text{net new}} = F_{\text{tugboat (new)}} - F_{\text{drag}} = 1,100,000 \text{ N} - 866,666.67 \text{ N} = 233,333.33 \text{ Newtons}
\][/tex]
3. Calculate the new acceleration ([tex]\( a_{\text{new}} \)[/tex]):
[tex]\[
a_{\text{new}} = \frac{F_{\text{net new}}}{m} = \frac{233,333.33 \text{ N}}{100,000,000 \text{ kg}} = 0.002333 \text{ m/s}^2
\][/tex]
4. Calculate the time to reach the speed of [tex]\( 0.2 \text{ m/s} \)[/tex] with the new force:
[tex]\[
v = a_{\text{new}} \times t_{\text{new}}
\][/tex]
[tex]\[
t_{\text{new}} = \frac{v}{a_{\text{new}}} = \frac{0.2 \text{ m/s}}{0.002333 \text{ m/s}^2} \approx 85.71 \text{ seconds}
\][/tex]
Thus, it would take approximately 85.71 seconds for the boat to reach the same speed if the new tugboat force is applied.
