The sports car has a mass of 2.3 Mg and accelerates at 6 m /...

The sports car has a mass of $2.3 \mathrm{Mg}$ and accelerates at $6 \mathrm{m} / \mathrm{s}^{2},$ starting from rest. If the drag resistance on the car due to the wind is $F_{D}=(10 v) \mathrm{N},$ where $v$ is the velocity in $\mathrm{m} / \mathrm{s},$ determine the power supplied to the engine when $t=5 \mathrm{s}$. The engine has a running efficiency of $\varepsilon=0.68$

Key Concepts

Efficiency in Energy Conversion

Efficiency in energy conversion relates the useful work output to the total energy input, accounting for energy losses. Here, the engine's efficiency indicates how much chemical or fuel energy is converted into mechanical energy, thereby adjusting the actual power supplied to the engine when calculating from the mechanical power output.

Instantaneous Mechanical Power

Instantaneous mechanical power is defined as the rate at which work is done, calculated as the product of force and velocity (P = Fv). This concept is crucial in determining the power output required from the engine at a specific moment when the car is accelerating against drag.

Constant Acceleration Kinematics

Constant acceleration kinematics involves equations that describe the motion of objects under a constant acceleration, such as v = at when starting from rest. This concept is needed to compute the car’s velocity at a given time, which in turn affects both the drag force and the power calculation.

Drag Force

Drag force is a resistive force acting opposite to the direction of motion, often dependent on the velocity of the moving object. In this problem, the drag resistance is given as F_D = (10v) N, highlighting how aerodynamic drag increases with speed and must be overcome by the engine for acceleration.

Newton's Second Law

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma). This principle is essential for determining the total force required to accelerate the car, taking into account both the force needed for acceleration and the opposing drag force.

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