Chapter 3 Acceleration and free fall - Light and Matter

Experiments also show that the acceleration is inversely proportional
to the object’s mass, and combining these two proportionalities
gives the following way of predicting the acceleration of any
object:
where
132 Chapter 4 Force and motion
Newton’s second law
a = Ftotal/m ,
m is an object’s mass
Ftotal is the sum of the forces acting on it, and
a is the acceleration of the object’s center of mass.
We are presently restricted to the case where the forces of interest
are parallel to the direction of motion.
An accelerating bus example 6
⊲ A VW bus with a mass of 2000 kg accelerates from 0 to 25 m/s
(freeway speed) in 34 s. Assuming the acceleration is constant,
what is the total force on the bus?
⊲ We solve Newton’s second law for F total = ma, and substitute
∆v/∆t for a, giving
F total = m∆v/∆t
= (2000 kg)(25 m/s − 0 m/s)/(34 s)
= 1.5 kN .
Some applications of calculus
Newton doesn’t care what frame of reference you use his laws
in, and this makes him different from Aristotle, who says there is
something special about the frame of reference attached firmly to
the dirt underfoot. Suppose that an object obeys Newton’s second
law in the dirt’s frame. It has some velocity that is a function of
time, and differentiating this function gives dv/dt = F/m. Suppose
we change to the frame of reference of a train that is in motion
relative to the dirt at constant velocity c. Looking out the window
of the train, we see the object moving with velocity v − c. But the
derivative of a constant is zero, so when we differentiate v − c, the
constant goes away, and we get exactly the same result. Newton is
still happy, although Aristotle feels a great disturbance in the force.