Does increasing your vehicle's mass increase your maneuverability on icy roads? Assume the vehicle and all contents are treated as a point mass.
It all comes down to two things: momentum and friction. Momentum (p) = the mass of the car (m) times the car's velocity (v), which is a term for a directed speed. We can change momentum by applying a force for a duration of time. Change in momentum = force × time.
Δp = F⋅t
Δm⋅v = F⋅t
What kind of force are we talking about? When it comes to driving a car, it's almost entirely related to the force of friction. When you apply the breaks, you're using
the friction between the tires and the road to stop the car. When you turn the wheel, it's the friction between the tires and the car that will ultimately change your direction, and when you accelerate, without the friction between the road and the tires, you'd just be spinning your wheels.
The force of friction (Ff) = the normal force (Fn) times the coefficient of friction (μ). The normal force of any object is equal to the product of the objects mass, the acceleration due to gravity, and the cosine of the angle (θ) between the horizontal and the surface providing the friction.
Ff = Fn⋅μ
Ff = m⋅g⋅cos(θ)⋅μ
Now, let's look at the change in momentum due to the force of friction applied over time t:
Δp = F⋅t
Δm⋅v = Ff⋅t
Δm⋅v = Fn⋅μ⋅t
Δm⋅v = m⋅g⋅cos(θ)⋅μ⋅t
The m on each side of the equation represents the same thing: the mass of the vehicle plus contents. Dividing each side by m gives us:
Δv = g⋅cos(θ)⋅μ⋅t
Thus, taken as a point mass, the mass of your vehicle has no effect on your ability to change your direction or speed.
Wednesday, March 19, 2008