Understanding the Problem
* Mass of the car: 1200 kg
* Initial velocity (u): 2.5 m/s
* Final velocity (v): 5.0 m/s
* Slope: 1 in 10 (meaning for every 10 meters traveled horizontally, the car rises 1 meter vertically)
* Distance traveled (s): 60 m
* Resistance force (R): 105 N
1. Energy Method
a) Calculate the work done against gravity:
* Vertical height (h): Since the slope is 1 in 10, the vertical rise for 60 meters traveled is (1/10) * 60 = 6 meters.
* Work done against gravity (Wg): Wg = mgh = 1200 kg * 9.8 m/s² * 6 m = 70560 J
b) Calculate the work done against resistance:
* Work done against resistance (Wr): Wr = R * s = 105 N * 60 m = 6300 J
c) Calculate the change in kinetic energy:
* Initial kinetic energy (KEi): KEi = (1/2) * m * u² = (1/2) * 1200 kg * (2.5 m/s)² = 3750 J
* Final kinetic energy (KEf): KEf = (1/2) * m * v² = (1/2) * 1200 kg * (5.0 m/s)² = 15000 J
* Change in kinetic energy (ΔKE): ΔKE = KEf - KEi = 15000 J - 3750 J = 11250 J
d) Calculate the total work done by the car:
* Total work (W): W = ΔKE + Wg + Wr = 11250 J + 70560 J + 6300 J = 88110 J
2. d'Alembert's Principle
a) Draw a free body diagram:
* Forces acting on the car:
* Gravity (mg) acting downwards
* Normal force (N) acting perpendicular to the slope
* Resistance force (R) acting opposite to the motion
* Driving force (F) acting parallel to the slope (this is what we're trying to find)
b) Apply d'Alembert's principle:
* Sum of forces = mass * acceleration
* F - mg sinθ - R = ma
c) Find the angle of the slope:
* sinθ: For a slope of 1 in 10, sinθ = (1/√(1² + 10²)) ≈ 0.0995
d) Find the acceleration:
* We can use the kinematic equation: v² = u² + 2as
* Solving for acceleration (a): a = (v² - u²) / (2s) = (5² - 2.5²) / (2 * 60) ≈ 0.2604 m/s²
e) Substitute and solve for the driving force (F):
* F = ma + mg sinθ + R
* F = (1200 kg * 0.2604 m/s²) + (1200 kg * 9.8 m/s² * 0.0995) + 105 N
* F ≈ 1955 N
Conclusion:
* Energy method: The total work done by the car is 88110 J.
* d'Alembert's principle: The driving force required is approximately 1955 N.
Note:
* The two methods give slightly different answers due to rounding errors and the fact that the energy method considers the work done against all forces, while d'Alembert's principle focuses on the net force.
* The driving force calculated using d'Alembert's principle is the force required to overcome the resistance, gravity, and to accelerate the car.
