(a) 9305 J
Let's start by finding the acceleration of the spelunker, through the following equation:
[tex]v^2-u^2=2ad[/tex]
where
v = 2.40 m/s is the final velocity
u = 0 is the initial velocity
a is the acceleration
d = 11.0 m is the distance covered
Solving for a,
[tex]a=\frac{v^2-u^2}{2d}=\frac{(2.40 m/s)^2-0}{2(11.0 m)}=0.26 m/s^2[/tex]
Now we can find the force lifting the spelunker. The equation for Newton's second law applied to the spelunker is:
[tex]F-mg = ma[/tex]
where
F is the lifting force
m = 84.0 kg is the mass of the spelunker
g = 9.81 m/s^2 is the acceleration due to gravity
a = 0.26 m/s^2 is the acceleration
Solving for F,
[tex]F=m(a+g)=(84.0 kg)(0.26 m/s^2+9.81 m/s^2)=845.9 N[/tex]
And now we can finally find the work done on the spelunker by the lifting force F:
[tex]W=Fd=(845.9 N)(11.0 m)=9305 J[/tex]
(b) 9064 J
In this case, the speed is constant, so the acceleration is zero. So Newton's second Law becomes
[tex]F-mg=0[/tex]
From which we find
[tex]F=mg=(84.0 kg)(9.81 m/s^2)=824.0 N[/tex]
And so the work done is
[tex]W=Fd=(824.0 N)(11.0 m)=9064 J[/tex]
(c) 8824 J
The acceleration of the spelunker here is given by
[tex]v^2-u^2=2ad[/tex]
where
v = 0 is the final velocity
u = 2.40 m/s is the initial velocity
a is the acceleration
d = 11.0 m is the distance covered
Solving for a,
[tex]a=\frac{v^2-u^2}{2d}=\frac{0-(2.40 m/s)^2}{2(11.0 m)}=-0.26 m/s^2[/tex]
Newton's second law applied to the spelunker is:
[tex]F-mg = ma[/tex]
where
F is the lifting force
m = 84.0 kg is the mass of the spelunker
g = 9.81 m/s^2 is the acceleration due to gravity
a = -0.26 m/s^2 is the acceleration
Solving for F,
[tex]F=m(a+g)=(84.0 kg)(-0.26 m/s^2+9.81 m/s^2)=802.2 N[/tex]
And now we can finally find the work done on the spelunker by the lifting force F:
[tex]W=Fd=(802.2 N)(11.0 m)=8824 J[/tex]
