Question

Power of an engine driving a vehicle of mass \(m\) with a speed \(v\) on a horizontal road is (\(\mu\) is the coefficient of friction between the road and the tyre)

• A. \(\frac{mg}{\mu v}\)
• B. \(\mu mgv\)
• C. \(\mu mgv^2\)
• D. \(\frac{\mu mg}{v}\)
• E. \(\frac{\mu mv}{g}\)

Hint:

For motion at constant speed, engine power is used only to overcome resistive forces like friction: \(P=Fv\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a vehicle to move at a constant speed, the driving force provided by its engine must be equal in magnitude and opposite in direction to the total resistive forces acting on it. In this case, the resistive force is the friction between the road and the tyres. Power is the rate at which the engine does work to provide this driving force.
Step 2: Key Formula or Approach:
1. Frictional Force: The force of kinetic friction (\(f_k\)) is given by \( f_k = \mu N \), where \( \mu \) is the coefficient of kinetic friction and N is the normal force.
2. Condition for Constant Velocity: For the vehicle to move at a constant velocity \(v\), the net force on it must be zero. This means the driving force from the engine (\(F_{engine}\)) must be equal to the frictional force (\(f_k\)).
3. Power: The power (\(P\)) delivered by a constant force (\(F\)) moving an object at a constant velocity (\(v\)) is given by the product of the force and velocity.
\[ P = F \cdot v \] Step 3: Detailed Explanation:
1. Find the Normal Force (N): The vehicle is on a horizontal road. The vertical forces are the weight (\(W = mg\)) acting downwards and the normal force (\(N\)) from the road acting upwards. Since there is no vertical acceleration, these forces balance.
\[ N = W = mg \] 2. Find the Frictional Force (\(f_k\)):
\[ f_k = \mu N = \mu mg \] 3. Find the Engine's Driving Force (\(F_{engine}\)): To maintain a constant speed \(v\), the engine's force must exactly oppose the friction.
\[ F_{engine} = f_k = \mu mg \] 4. Calculate the Power (P): The power delivered by the engine is the product of its driving force and the speed of the vehicle.
\[ P = F_{engine} \times v \] \[ P = (\mu mg) \times v = \mu mgv \] Step 4: Final Answer:
The power of the engine is \( \mu mgv \). This corresponds to option (B).