a/Power and force are the
rates at which energy and
momentum are transferred.b/The airbag increases ∆t
so as to reduceF=∆p/∆t.3.2 Force in one dimension
3.2.1 Momentum transfer
For every conserved quantity, we can define an associated rate
of flow. An open system can have mass transferred in or out of it,
and we can measure the rate of mass flow, dm/dtin units of kg/s.
Energy can flow in or out, and the rate of energy transfer is the
power,P = dE/dt, measured in watts.^3 The rate ofmomentum
transfer is called force,F=
dp
dt[definition of force].The units of force are kg·m/s^2 , which can be abbreviated as newtons,
1 N = kg·m/s^2. Newtons are unfortunately not as familiar as watts.
A newton is about how much force you’d use to pet a dog. The
most powerful rocket engine ever built, the first stage of the Saturn
V that sent astronauts to the moon, had a thrust of about 30 million
newtons. In one dimension, positive and negative signs indicate the
direction of the force — a positive force is one that pushes or pulls
in the direction of the positivexaxis.
Walking into a lamppost example 20
.Starting from rest, you begin walking, bringing your momentum
up to 100 kg·m/s. You walk straight into a lamppost. Why is the
momentum change of−100 kg·m/s so much more painful than
the change of +100 kg·m/s when you started walking?
.The forces are not really constant, but for this type of qualitative
discussion we can pretend they are, and approximate dp/dtas
∆p/∆t. It probably takes you about 1 s to speed up initially, so the
ground’s force on you isF=∆p/∆t≈100 N. Your impact with
the lamppost, however, is over in the blink of an eye, say 1/10 s or
less. Dividing by this much smaller∆tgives a much larger force,
perhaps thousands of newtons (with a negative sign).
This is also the principle of airbags in cars. The time required
for the airbag to decelerate your head is fairly long: the time it
takes your face to travel 20 or 30 cm. Without an airbag, your face
would have hit the dashboard, and the time interval would have
been the much shorter time taken by your skull to move a couple of
centimeters while your face compressed. Note that either way, the
same amount of momentum is transferred: the entire momentum of
your head.
Force is defined as a derivative, and the derivative of a sum is
the sum of the derivatives. Therefore force is additive: when more
than one force acts on an object, you add the forces to find out
what happens. An important special case is that forces can cancel.
Consider your body sitting in a chair as you read this book. Let
(^3) Recall that uppercasePis power, while lowercasepis momentum.
Section 3.2 Force in one dimension 149