Department of Physics United States Naval Academy Lecture 22

Home , Closed system, Collision, Elastic collision, Inelastic collision

Department of Physics United States Naval Academy Lecture 22: Collision and Conservation of Linear Momentum Learning Objectives • Apply the conservation of linear momenta to relate the initial momenta of the particles to their momenta at a later instant.

• Distinguish between elastic collisions, inelastic collisions, and completely inelastic collisions.

Conservation of Linear Momentum: If a system is closed (no external forces) and isolated (no mass enters or leaves) so that no net external force acts on it, then the linear momentum ~P must be constant even if there are internal changes. In order words, the the total momentum of an isolated and closed system is conserved; i.e., remains constant with time. ~P = constant

This conservation of linear momentum (in isolated and closed systems) can also be written in terms of the system’s initial momentum and its momentum at some later instant using Newton’s second law; ∆~P ~F = = 0 ⇒ ∆~P = 0 ⇒ ~P = ~P ext ∆t i f If the component of the net external force on a closed system is zero along an axis, then the component of the linear momentum of the system along that axis cannot change.

Collision in One Dimension: According to Newton’s second law, the momentum ~p of any particle-like body cannot change unless a net external force changes it. This is commonly achieved via collisions. In a typical collision (or crash), the external force on the body is brief, has large magnitude (impulse), and suddenly changes the body’s momentum. Since momentum is conserved in all collisions, we can use the kinetic energy to distinguish between the two main types of collision.

(a) Inelastic collisions: In an inelastic collision of two bodies, the kinetic energy of the two-body system is not conserved. If the system is closed and isolated, the total linear momentum of the system must be conserved, which we can write in vector form as

~P1i +~P2i = ~P1 f +~P2 f

where subscripts i and f refer to values just before and just after the collision, respectively.

* Perfectly inelastic: If the bodies stick together, the collision is a completely (perfectly) inelastic collision and the bodies have the same ﬁnal velocity V (because they are stuck together).

The center of mass of a closed, isolated system of two colliding bodies is not aﬀected by a collision. In particular, the

velocity ~vcom of the center of mass cannot be changed by the collision.

(b) Elastic collisions: An elastic collision is a special type of collision in which the kinetic energy of a system of colliding bodies is conserved. If the system is closed and isolated, its linear momentum is also conserved.

Collisions in Two Dimensions: If two bodies collide and their motion is not along a single axis (the collision is not head-on), the collision is two-dimensional. If the two-body system is closed and isolated, the law of conservation of momentum applies to the collision and can be written as

P~1i + P~2i = P~1 f + P~2 f

In component form, the law gives two equations that describe the collision (one equation for each of the two dimensions). If the collision is also elastic (a special case), the conservation of kinetic energy during the collision gives a third equation:

K1i + K2i = K1 f + K2 f

A steel ball of mass 0.500 kg is fastened to a cord that is 70.0 cm long and ﬁxed at the far end. The ball is then released when the cord is horizontal as shown. At the bottom of its path, the ball strikes a 2.50 kg steel block initially at rest on a frictionless surface. The collision is elastic. Find (a) the speed of the ball and (b) the speed of the block, both just after the collision.

A projectile proton with a speed of 500 m/s collides elastically with a target proton initially at rest. The two protons then move along perpendicular paths, with the projectile path at 60◦ from the original direction. After the collision, what are the speeds of (a) the target proton and (b) the projectile proton?

A 59kg person sits with a large box of mass 24 kg at the center of a frictionless frozen pond. To get to the edge of the pond, the person pushes oﬀ of the box attaining a velocity of 0.85 m/s East. What is the velocity of the box after the push?

A horizontally ﬁred .012kg bullet embeds into a 2.4kg ballistic pendulum. The “bullet + pendulum” swings rising 0.12m vertically. What was v0 of the bullet?

Block A of mass 5m moves along the −x axis approaching the origin with vx = +v0. It collides elastically with block B of mass 3m that lies at rest at the origin. What is the velocity of block B after the collision?

ˆ ˆ ˆ Clay lump A of mass 1.8 kg has velocity~vA = 2.5i m/s. Clay lump B of mass 3.7 kg has velocity~vB = (−.75i+1.7 j) m/s. The lumps collide and stick together.

• What is the ﬁnal velocity of the big lump?

• What percentage of this system’s initial kinetic energy converted into thermal energy?