Answer:
a) The average friction force exerted on the toboggan is 653.125 newtons, b) The rough region reduced the kinetic energy of the toboggan in 92.889 %, c) The speed of the toboggan is reduced in 73.333 %.
Explanation:
a) Given the existence of non-conservative forces (friction between toboggan and ground), the motion must be modelled by means of the Principle of Energy Conservation and the Work-Energy Theorem, since toboggan decrease its speed (associated with due to the action of friction. Changes in gravitational potential energy can be neglected due to the inclination of the ground. Then:
Where:
, 
are the initial and final translational kinetic energies of the tobbogan, measured in joules.
- Dissipated work due to friction, measured in joules.
By applying definitions of translation kinetic energy and work, the expression described above is now expanded and simplified:
Where:
- Friction force, measured in newtons.
- Distance travelled by the toboggan in the rough region, measured in meters.
- Mass of the toboggan, measured in kilograms.
, 
- Initial and final speed of the toboggan, measured in meters per second.
The friction force is cleared:
If 
, 
, 
and 
, then:
![f = \frac{(375\,kg)\cdot \left[\left(4.50\,\frac{m}{s} \right)^{2}-\left(1.20\,\frac{m}{s}\right)^{2}\right]}{2\cdot (5.40\,m)}](https://tex.z-dn.net/?f=f%20%3D%20%5Cfrac%7B%28375%5C%2Ckg%29%5Ccdot%20%5Cleft%5B%5Cleft%284.50%5C%2C%5Cfrac%7Bm%7D%7Bs%7D%20%5Cright%29%5E%7B2%7D-%5Cleft%281.20%5C%2C%5Cfrac%7Bm%7D%7Bs%7D%5Cright%29%5E%7B2%7D%5Cright%5D%7D%7B2%5Ccdot%20%285.40%5C%2Cm%29%7D)
The average friction force exerted on the toboggan is 653.125 newtons.
b) The percentage lost by the kinetic energy of the tobbogan due to friction is given by the following expression, which is expanded and simplified afterwards:
![\%K_{loss} = \left[1-\left(\frac{v_{2}}{v_{1}}\right)^{2} \right]\times 100\,\%](https://tex.z-dn.net/?f=%5C%25K_%7Bloss%7D%20%3D%20%5Cleft%5B1-%5Cleft%28%5Cfrac%7Bv_%7B2%7D%7D%7Bv_%7B1%7D%7D%5Cright%29%5E%7B2%7D%20%5Cright%5D%5Ctimes%20100%5C%2C%5C%25)
If and , then:
![\%K_{loss} = \left[1-\left(\frac{1.20\,\frac{m}{s} }{4.50\,\frac{m}{s} }\right)^{2} \right]\times 100\,\%](https://tex.z-dn.net/?f=%5C%25K_%7Bloss%7D%20%3D%20%5Cleft%5B1-%5Cleft%28%5Cfrac%7B1.20%5C%2C%5Cfrac%7Bm%7D%7Bs%7D%20%7D%7B4.50%5C%2C%5Cfrac%7Bm%7D%7Bs%7D%20%7D%5Cright%29%5E%7B2%7D%20%5Cright%5D%5Ctimes%20100%5C%2C%5C%25)
The rough region reduced the kinetic energy of the toboggan in 92.889 %.
c) The percentage lost by the speed of the tobbogan due to friction is given by the following expression:
If and , then:
The speed of the toboggan is reduced in 73.333 %.
