(a) [tex]4.3\cdot 10^9 J[/tex]
The kinetic energy of an object is given by:
[tex]K=\frac{1}{2}mv^2[/tex]
where
m is the mass of the object
v is its speed
For the ship in the problem, we have
[tex]m=6.40\cdot 10^7 kg[/tex] is the mass
[tex]v=11.6 m/s[/tex] is the speed
So its kinetic energy is
[tex]K=\frac{1}{2}(6.40\cdot 10^7 kg)(11.6 m/s)^2=4.3\cdot 10^9 J[/tex]
(b) [tex]-4.3\cdot 10^9 J[/tex]
According to the work-kinetic energy theorem, the work done on an object is equal to the change in kinetic energy of the object:
[tex]W= \Delta K = \frac{1}{2}mv^2 - \frac{1}{2}mu^2[/tex]
where
W is the work done
m is the mass of the object
v is the final speed of the object
u is the initial speed of the object
Here we want to find the work done to stop the ship, so the final speed of the ship is v=0, while the initial speed is u=11.6 m/s. So the work done will be
[tex]W= 0 -\frac{1}{2}(6.40\cdot 10^7 kg)(11.6 m/s)^2=-4.3\cdot 10^9 J[/tex]
(c) [tex]1.3\cdot 10^6 N[/tex]
The work done on an object can be also written as follows
[tex]W=Fd[/tex]
where
F is the magnitude of the force
d is the displacement of the object
Here we know:
[tex]W=-4.3\cdot 10^9 J[/tex] is the work done
d = 3.20 km = 3200 m is the displacement of the ship
So we can solve the formula to find F, the force exerted on the ship to stop it:
[tex]F=\frac{W}{d}=\frac{-4.3\cdot 10^9 J}{3200 m}=-1.3\cdot 10^6 N[/tex]
and the negative sign simply means that the force is opposite to the displacement of the ship (in fact, the force acts against the motion of the ship).
