Download Review of Momentum The Momentum Principle

Momentum Principle
Objective: Define the momentum principle both verbally and mathematically; know how to calculate net
force if given various force vectors acting on an object; know how to calculate the change in momentum for
an object during some time interval.
Review of Momentum
It’s important for you to be able to calculate a momentum vector if given mass and velocity. Please be sure
that you can use the following equation (or its simplified form p~ = m~v for nonrelativistic speeds).
m~v
momentum p~ = q
1−
|~
v |2
c2
(1)
The Momentum Principle
The momentum principle governs all motion except for subatomic and nuclear interactions (use quantum
mechanics for that) and interactions involving very massive objects (use general theory of relativity for that).
It relates the “cause of motion”–interactions–to the “effect” of those interactions–motion.
If we refer to an interaction as a force–which is just a quantitative way to measure an interaction–and if
we refer the net result of lots of interactions as the net force, then the momentum principle states that
For a short time interval, the change in momentum is equal to the net force times the duration
of interaction.
This is more familiarly known as Newton’s second law. It is expressed mathematically as
∆~
p = F~net ∆t for a small time interval
(2)
Let’s see if this is consistent with Newton’s first law which states that if there are no interactions or if all
of the interactions cancel each other’s effects, then an object will have uniform motion (constant direction
and speed).If the net force on an object is zero, then its change in momentum is zero. That can only happen
if the object has a constant velocity–it is indeed consistent with Newton’s first law.
The unit of force is the newton (N) which is a kg m/s2 .
Predicting a new momentum
Let’s say that you know the momentum of a GPS satellite at some instant of time, and you want to know
where it will be at some small time interval later if you fire its thrusters in a particular direction. How would
you calculate it? You would use the momentum principle. The new momentum will be
p~new = p~old + F~net ∆t for a small time interval
(3)
where the new momentum is sometimes called the “initial” momentum and the old momentum is sometimes
called the “final” momentum. Really, they are just the momentum at the end and beginning of the time
interval.
Calculus
The momentum principle can be used even a long time interval if, the net force is the average net force
during the time interval. However, we often know the net force at an instant of time. If this is the case,
then our calculations will only be correct (approximately) for a small time interval.
As long as we use VERY small time intervals, the momentum principle is exactly correct for an instantaneous net force. So, take the limit as ∆t approaches zero, and
d~
p
F~net =
dt
This is exactly correct for the net force at an instant of time.
(4)
Impulse
There is a name for the quantity F~net ∆t–it is called net impulse. Impulse tells you something about the net
force and the duration of the net force.
The new ion engine that NASA has developed and tested (see http://www.spaceref.com/news/viewpr.html?pid=12209)
exerts a fairly small force on a space probe (compared to conventional fuel thrusters). However, it can exert
this force for a VERY long time interval, thus resulting in a much greater impulse on the space probe. If
we ever hope to send a probe to Pluto (note that it would take a VERY long time to get there), it would
probably use an ion engine to achieve greater speeds.
Application
1. A baseball (m=0.155 kg) is moving at a speed of 40 m/s in the direction ¡-1,0,0¿ when it is hit by a
bat. It leaves the bat 0.002 s later at a speed of 30 m/s in the direction ¡0.333, 0.667, 0.667¿. What
was the change in momentum of the baseball during the time interval it was in contact with the bat?
What was the net force on the baseball while it was in contact with the bat? What was the net impulse
on the ball while it was in contact with the bat?
2. Suppose two physics students get into a heated discussion regarding the net force on the baseball while
it is in contact with the bat (in the previous question). Student A says that the net force on the
baseball is just the force of the bat on the ball. Student B says that the net force on the baseball is
the net result of the force of the bat on the ball and the gravitational force of the earth on the ball.
Who is right?
3. Sketch the change in momentum vector for a 1000 kg car that is moving at a speed of 20 m/s in the
direction ¡1,0,0¿ and after a collision is moving at 10 m/s in the direction ¡0.709,-0.709¿. What was
the direction of the net force on the car during the collision?
4. A tennis ball (m=0.057 kg) tossed upward by the great High Point University tennis star, Neil Hammel,
is moving upward at a speed of 0.5 m/s at some instant. A short time later, after it’s reached its peak
and is falling downward, it is moving downward at 0.5 m/s. Sketch the change in momentum of the
tennis ball. What is the direction of the net force on the ball?
5. Suppose that a conventional rocket engine can exert a constant net force of magnitude 1.0 × 105 N for
100 hours. Suppose an ion engine can exert a constant net force of magnitude 10 N for 1.0 × 101 0 hours
(I’m making these numbers up). If probe A uses a conventional rocket and probe B uses an ion engine,
which will reach the greater speed if they both start from rest and experience no other interactions?