(1) 8408 J = 8.4 kJ
We need to find the acceleration of the spelunker first, through the equation:
[tex]v^2-u^2=2ad[/tex]
where
v = 4.40 m/s is the final speed
u = 0 is the initial speed
a is the acceleration
d = 10.0 m is the distance
Solving the equation for a, we find
[tex]a=\frac{v^2-u^2}{2d}=\frac{(4.40 m/s)^2-0}{2(10.0 m)}=0.97 m/s^2[/tex]
In order to find the magnitude of force F used to lift the spelunker, we have to apply Newton's second law:
[tex]\sum F = ma\\F - mg = ma[/tex]
where (mg) is the weight of the spelunker, and a = 0.97 m/s^2. Solving for F, we find
[tex]F=m(g+a)=(78.0 kg)(9.81 m/s^2+0.97 m/s^2)=840.8 N[/tex]
And so, the work done by the force during this stage is
[tex]W=Fd=(840.8 N)(10.0 m)=8408 J[/tex]
(2) 7644 J = 7.6 kJ
The work done on the spelunker in this stage is
[tex]W=Fd[/tex]
where F is the force applied on the spelunker to lift him, and d = 10.0 m is the vertical distance through which the spelunker is lifted.
In order to find the magnitude of F, we have to apply Newton's second law:
[tex]\sum F = ma\\F - mg = ma[/tex]
where (mg) is the weight of the spelunker, and the acceleration is zero because he is moving at constant speed: so, a=0, and the equation becomes
[tex]F-mg=0\\F=mg=(78.0 kg)(9.8 m/s^2)=764.4 N[/tex]
So, the work done is
[tex]W=(764.4 N)(10.0 m)=7644 J[/tex]
3) 6895 J = 6.9 kJ
This stage is similar to stage (1); we find the deceleration using:
[tex]v^2-u^2=2ad[/tex]
where
v = 0 m/s is the final speed
u = 4.40 is the initial speed
a is the acceleration
d = 10.0 m is the distance
Solving the equation for a, we find
[tex]a=\frac{v^2-u^2}{2d}=\frac{0-(4.40 m/s)^2}{2(10.0 m)}=-0.97 m/s^2[/tex]
In order to find the magnitude of force F used to lift the spelunker, we have to apply Newton's second law:
[tex]\sum F = ma\\F - mg = ma[/tex]
where (mg) is the weight of the spelunker, and a = -0.97 m/s^2. Solving for F, we find
[tex]F=m(g+a)=(78.0 kg)(9.81 m/s^2-0.97 m/s^2)=689.5 N[/tex]
And so, the work done by the force during this stage is
[tex]W=Fd=(689.5 N)(10.0 m)=6895 J[/tex]
