Unit 4: Linear Momentum

17 terms. Say what each one means, then open the card to check.

• Momentum

  The product of an object’s mass and velocity, $\vec{p} = m\vec{v}$. It is a vector that points the way the object moves, measured in kg·m/s.

  Example. A 1,000 kg boat drifting east at 0.50 m/s has a momentum of 500 kg·m/s east.

• Impulse

  The average force on an object times the time it acts, $J = F\Delta t$, in N·s. It points the same way as the force.

  Example. A steady 15 N push lasting 0.20 s gives an impulse of 3.0 N·s in the direction of the push.

• Impulse-momentum theorem

  The net impulse on an object equals its change in momentum: $F_{\text{net}}\Delta t = \Delta p$. One N·s is the same as one kg·m/s.

  Example. A net impulse of 6.0 N·s on a 2.0 kg cart at rest leaves it moving at 3.0 m/s.

• Area under a force-time graph

  When the force changes, the impulse is the area between the force-time graph and the time axis; area below the axis counts as negative impulse.

  Example. A force that rises steadily from 0 to 10 N over 0.40 s gives an impulse of one-half times 10 N times 0.40 s, or 2.0 N·s.

• Change in momentum

  Final momentum minus initial momentum, with signs for direction. An object that reverses direction changes its momentum more than one that only stops.

  Example. A 2.0 kg ball at +3.0 m/s that bounces back at -3.0 m/s has $\Delta p = -12$ kg·m/s; if it had stopped, $\Delta p$ would be -6.0 kg·m/s.

• Contact time and average force

  For a fixed change in momentum, a longer contact time means a smaller average force, because $F_{\text{avg}} = \Delta p/\Delta t$.

  Example. A gymnast who lands on a thick mat takes longer to stop than on a bare floor, so the average force on her legs is smaller.

• System

  The object or group of objects you choose to study. Forces between objects inside it are internal; forces from anything outside it are external.

  Example. For two colliding pucks taken together, their pushes on each other are internal, but friction from the ice is external.

• Conservation of momentum

  If the net external force on a system is zero, the total momentum of the system stays the same, even though the momenta of its parts change.

  Example. Two carts held together by a compressed spring and released from rest move apart with equal and opposite momenta, so the total stays zero.

• Equal and opposite impulses

  In a collision the two objects push on each other with equal and opposite forces for the same time, so their changes in momentum are equal in size and opposite in direction.

  Example. When a bus and a bicycle collide, the bicycle’s change in momentum is the same size as the bus’s, though its change in velocity is far larger.

• Center-of-mass velocity

  The total momentum divided by the total mass, $v_{\text{cm}} = (m_1v_1 + m_2v_2)/(m_1 + m_2)$. It stays constant when no net external force acts on the system.

  Example. A 3.0 kg block at +2.0 m/s and a 1.0 kg block at rest have $v_{\text{cm}} = +1.5$ m/s, before and after they collide.

• Recoil

  When parts of a system at rest push apart, they move off in opposite directions with momenta of equal size, so the lighter part moves faster.

  Example. A 2.0 kg toy launcher that fires a 0.10 kg foam ball forward at 8.0 m/s on a frictionless floor rolls back at 0.40 m/s.

• Momentum in two dimensions

  Momentum is conserved separately along each of two perpendicular directions, so you can write one conservation equation for x and another for y.

  Example. If two pucks have no total north-south momentum before they collide, they have none afterward, whatever happens east-west.

• Elastic collision

  A collision in which the total kinetic energy of the system is the same before and after, along with the total momentum.

  Example. Two hardened steel ball bearings that bounce apart with almost no heating or sound come very close to an elastic collision.

• Inelastic collision

  A collision in which total momentum is conserved but some kinetic energy becomes other forms, such as internal energy, sound or permanent deformation.

  Example. A bumper that crumples in a crash turns part of the cars’ kinetic energy into bent metal and heating.

• Perfectly inelastic collision

  An inelastic collision in which the objects stick together and share one final velocity; it loses the most kinetic energy that momentum conservation allows.

  Example. A 0.20 kg beanbag at 5.0 m/s that lands in and stays in a 0.80 kg cart at rest leaves both moving at 1.0 m/s.

• Separation speed in elastic collisions

  In a head-on elastic collision, the objects move apart from each other at the same relative speed at which they came together.

  Example. Two gliders that close in on each other at a combined 1.6 m/s and collide elastically move apart at 1.6 m/s relative to each other.

• Equal masses, elastic and head-on

  When two objects of equal mass collide head-on and elastically, they swap velocities.

  Example. A croquet ball that strikes an identical ball at rest head-on nearly stops, and the struck ball rolls off at almost the first ball’s speed.