The velocity 'V' reached by a car of mass 'm' at certain distance from the starting point driven with constant power 'P' is such that

Explanation

Here's how we can find the relationship between the velocity (V), mass (m), power (P), and distance (x):1. Work-Energy Principle:We know that the work done (W) by the engine is equal to the change in kinetic energy (ΔKE) of the car:W = ΔKE2. Work Done by the Engine:The work done by the engine (W) can be calculated using the following equation:W = P * twhere:P is the constant powert is the time taken to travel the distance3. Change in Kinetic Energy:The change in kinetic energy (ΔKE) can be calculated using the following equation:ΔKE = 1/2 * m * V^2where:m is the mass of the carV is the final velocity of the car4. Relating Distance and Time:Since the power is constant, we can relate the distance (x) traveled by the car to the time (t) using the following equation:P = F * Vwhere:F is the force acting on the carV is the instantaneous velocity of the carAs the car accelerates, the velocity (V) increases. However, for a small distance increment (dx), we can assume the velocity to be constant. Therefore, we can write:dx = V * dtIntegrating both sides over the entire distance (x):x = ∫ V * dt = ∫ (P / F) * dtSince power (P) and mass (m) are constant, we can take them outside the integral:x = (P / m) * ∫ 1/V * dt5. Combining Equations:Substituting the expressions for work (W) and change in kinetic energy (ΔKE) from equations 2 and 3 into the work-energy principle (equation 1):P * t = 1/2 * m * V^2Substituting the expression for time (t) from equation 4:P * ((x * m) / (P)) = 1/2 * m * V^2Simplifying:2x = V^26. Solving for Velocity:Taking the square root of both sides:V = √(2x)Cubing both sides:V^3 = 2^3 * x = 8xNow, substituting the expression for x from equation 4:V^3 = 8 * (P / m)Multiplying both sides by the constant α (which represents a proportionality constant):V^3 * α = 8α * (P / m)Therefore, the correct relationship between the velocity (V), mass (m), power (P), and distance (x) is V^3 * α * 3P / m, which corresponds to option (c).