Resistance to the motion of an automobile consists of road...

Resistance to the motion of an automobile consists of road friction, which is almost independent of speed, and air drag, which is proportional to speed-squared. For a certain car with a weight of $12000 \mathrm{~N}$, the total resistant force $F$ is given by $F=300+1.8 v^{2}$, with $F$ in newtons and $v$ in meters per second. Calculate the power (in horsepower) required to accelerate the car at $0.92 \mathrm{~m} / \mathrm{s}^{2}$ when the speed is $80 \mathrm{~km} / \mathrm{h}$.

Key Concepts

Newton's Laws of Motion

Newton's Second Law, which states that the net force on an object equals its mass times its acceleration (F = ma), is fundamental in determining the force necessary to accelerate an object. This principle is applied to calculate how much additional force must be provided on top of any resistive forces for a vehicle to increase its speed.

Resistive Forces

Resistive forces in vehicle dynamics include frictional forces and aerodynamic drag. Frictional forces, such as road friction, tend to be more or less constant regardless of speed, while aerodynamic drag increases with the square of the speed. Understanding the nature of these forces is essential for analyzing the energy and power required to overcome them.

Power in Mechanics

Power is the rate at which work is performed or energy is transferred, and it is calculated as the product of force and velocity (P = F × v). In the context of vehicle dynamics, power quantifies the energy required per unit time to both overcome resistive forces and achieve a desired acceleration.

Unit Conversion and Consistency

Ensuring unit consistency is critical in solving problems in mechanics. This includes converting speed units, such as from kilometers per hour to meters per second, and converting power units, for example from watts to horsepower, to maintain the validity of calculations and to express the final answer in the required units.

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