c/Isaac Newton (1643-1727).
the positivexaxis be upward. The chair’s upward force on you
is represented with a positive number, which cancels out with the
earth’s downward gravitational force, which is negative. The total
rate of momentum transfer into your body is zero, and your body
doesn’t change its momentum.
Finding momentum from force example 21
.An object of massmstarts at rest att=to. A force varying as
F =bt−^2 , wherebis a constant, begins acting on it. Find the
greatest speed it will ever have.
.F=
dp
dt
dp=Fdtp=∫
Fdt+po=−
b
t
+po,wherepois a constant of integration. The given initial condition
is thatp= 0 att =to, so we find thatpo=b/to. The negative
term gets closer to zero with increasing time, so the maximum
momentum is achieved by lettingtapproach infinity. That is, the
object will never stop speeding up, but it will also never surpass a
certain speed. In the limitt→ ∞, we identifypoas the momen-
tum that the object will approach asymptotically. The maximum
velocity isv=po/m=b/mto.
Discussion Question
A Many collisions, like the collision of a bat with a baseball, appear to
be instantaneous. Most people also would not imagine the bat and ball as
bending or being compressed during the collision. Consider the following
possibilities:
(1) The collision is instantaneous.
(2) The collision takes a finite amount of time, during which the ball and
bat retain their shapes and remain in contact.
(3) The collision takes a finite amount of time, during which the ball and
bat are bending or being compressed.
How can two of these be ruled out based on energy or momentum con-
siderations?3.2.2 Newton’s laws
Although momentum is the third conserved quantity we’ve en-
countered, historically it was the first to be discovered. Isaac New-
ton formulated a complete treatment of mechanical systems in terms
of force and momentum. Newton’s theory was based on three laws
of motion, which we now think of as consequences of conservation
of mass, energy, and momentum.150 Chapter 3 Conservation of Momentum