91Ó°ÊÓ

We can use the conservation of mechanical energy principle. The initial kinetic energy of the car will be converted into potential energy as it goes up the hill. We have the initial kinetic energy: \(K_{i} = \frac{1}{2}mv_{i}^2\) and the final potential energy: \(U_{f} = mgh\) Since there is no friction, we have: \(K_{i} = U_{f}\) Now we need to find the initial speed in meters per second, and then we can find the height: \(v_{i} = 110 \frac{\mathrm{km}}{\mathrm{h}} \times \frac{1000 \mathrm{m}}{\mathrm{km}} \times \frac{1 \mathrm{h}}{3600 \mathrm{s}} = 30.56 \frac{\mathrm{m}}{\mathrm{s}}\) Write the equation to find the height the car can coast up: \(h = \frac{1}{2}\frac{v_{i}^2}{g}\)

Now, we plug in the initial speed of the car and the acceleration due to gravity (\(g = 9.81 \frac{\mathrm{m}}{\mathrm{s}^2}\)): \(h = \frac{1}{2}\frac{(30.56 \frac{\mathrm{m}}{\mathrm{s}})^2}{9.81 \frac{\mathrm{m}}{\mathrm{s}^2}}\) \(h = 47.63 \mathrm{m}\) The height the car can coast up without friction is 47.63 m.

We need to find the thermal energy generated by friction. First, we'll calculate the initial and final mechanical energies, and use work-energy principle: \(W_{nc} = E_f - E_i\) Where \(W_{nc}\) represents the work done by non-conservative forces (friction), and \(E_i\) and \(E_f\) are the initial and final mechanical energies, respectively. The initial mechanical energy is the initial kinetic energy: \(E_i = K_{i} = \frac{1}{2}mv_{i}^2\) The final mechanical energy is the final potential energy (when the car is at the height of 22.0 m): \(E_f = U_{f} = mgh_f\) Now, plug in the mass, initial speed, acceleration due to gravity, and the final height: \(W_{nc} = (750\mathrm{kg})(9.81\frac{\mathrm{m}}{\mathrm{s}^2})(22.0\mathrm{m}) - \frac{1}{2}(750\mathrm{kg})(30.56\frac{\mathrm{m}}{\mathrm{s}})^2\) \(W_{nc} = -261720 \mathrm{J}\) The negative sign indicates that this work was done against the car's motion. The energy generated by friction is 261720 J.

To find the average friction force, we need to find the distance the car traveled up the slope and then use the work done by friction: \(F_{avg} = \frac{W_{nc}}{d}\) We can use trigonometry to find the distance: \(d = \frac{h_f}{\sin(\theta)}\) Plug in the final height and the slope angle (converted to radians): \(d = \frac{22.0\mathrm{m}}{\sin(2.5^{\circ})}\) \(d = 505.44 \mathrm{m}\) Now, we can find the average friction force: \(F_{avg} = \frac{-261720 \mathrm{J}}{505.44 \mathrm{m}}\) \(F_{avg} = -517.62 \mathrm{N}\) The average force of friction is 517.62 N, opposing the car's motion up the hill.