# Essay on linear equations

Rockets also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket. 13 Multiple dimensions edit Two-dimensional elastic collision. There is no motion perpendicular to the image, so only two components are needed to represent the velocities and momenta. The two blue vectors represent velocities after the collision and add vectorially to get the initial (red) velocity. Real motion has both direction and velocity and must be represented by a vector. In a coordinate system with x, y, z axes, velocity has components v x in the x -direction, v y in the y -direction, v z in the z -direction. The vector is represented by a boldface symbol: 14 v(vx,vy, vz).displaystyle mathbf v left(v_x,v_y,v_zright).

If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change. Inelastic collisions edit main article: Inelastic collision a perfectly inelastic collision between equal masses In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as heat or sound ). Examples include traffic collisions, 10 in which the effect of lost kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the FranckHertz experiment 11 and particle accelerators in which the kinetic energy. In a perfectly inelastic collision (such as a bug hitting a windshield both bodies have the same motion afterwards. If one body is motionless to begin with, the equation for conservation of momentum is m1u1(m1m2)v,displaystyle m_1u_1left(m_1m_2right)v so vm1m1m2u1.displaystyle vfrac m_1m_1m_2u_1. In a frame of reference moving at the speed v the objects are brought to rest by the collision and 100 of the kinetic energy is converted to other forms of energy. One measure of the inelasticity of the collision is the coefficient of restitution c r, defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula: 12 CRbounce heightdrop height. Displaystyle C_textRsqrt frac textbounce heighttextdrop height. The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that transforms potential energy stored resume in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation.

For example, suppose there are two bodies of equal mass m, one stationary and one approaching the other at a speed beauty v (as in the figure). The center of mass is moving at speed v /2 and both bodies are moving towards it at speed v /2. Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed. The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by 4 v1u2v2u1.displaystyle in general, when the initial velocities are known, the final velocities are given by 9 v_1left(frac 2m_2m_1m_2right)u_2, v_2left(frac 2m_1m_1m_2right)u_1.

This is not necessarily conserved. If it is conserved, the collision is called an elastic collision ; if not, it is an inelastic collision. Elastic collisions edit main article: Elastic collision Elastic collision of equal masses Elastic collision of unequal masses An elastic collision is one in which no kinetic energy is absorbed in the collision. Perfectly elastic "collisions" can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps them apart. A slingshot maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two pool balls is a good example of an almost totally elastic collision, due to their high rigidity, but when bodies come in contact there is always some dissipation. 8 A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are u 1 and u 2 before the collision and v review 1 and v 2 after, the equations expressing conservation of momentum and kinetic energy are: 12m_1u_12tfrac 12m_2u_22 tfrac 12m_1v_12tfrac 12m_2v_22,.endaligned A change of reference frame can simplify analysis of a collision.

Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or Galilean invariance. 7 A change of reference frame, can, often, simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest. Another, commonly used reference frame, is the center of mass frame one that is moving with the center of mass. In this frame, the total momentum is zero. Application to collisions edit by itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision. Another property of the motion, kinetic energy, must be known.

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In the migration elevator's and person B's frames of reference, it has zero velocity and momentum. Momentum is a measurable quantity, and the measurement depends on the motion of the observer. For example: if an apple is sitting in a glass elevator that is descending, an outside observer, looking into the elevator, sees the apple moving, so, to that observer, the apple has a non-zero momentum. To someone inside the elevator, the apple does not move, so, it has zero momentum. The two observers each have a frame of reference, in which, they observe motions, and, if the elevator is descending steadily, they will see behavior that is consistent with those same physical laws. Suppose a particle has position x in a stationary frame of reference. From the point of view of another frame of reference, moving at a uniform speed u, the position (represented by a primed coordinate) changes with time as xxut.

This is called a galilean transformation. If the particle is moving at speed dx / dt v in the first frame of reference, in the second, it is moving at speed vdxdtvu. Since u does not change, the accelerations are the same: advdta. Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, newton's second law is unchanged.

The rate of change of momentum is 3 (kgm/s s. Conservation edit In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum is constant. This fact, known as the law of conservation of momentum, is implied by newton's laws of motion. 4 5 Suppose, for example, that two particles interact. Because of the third law, the forces between them are equal and opposite. If the particles are numbered 1 and 2, the second law states that F 1 dp 1/ dt and F 2 dp 2/.

Therefore, dp1dtdp2dt, displaystyle frac dp_1dt-frac dp_2dt, with the negative sign indicating that the forces oppose. Equivalently, ddt(p1p2)0.displaystyle frac ddtleft(p_1p_2right)0. If the velocities of the particles are u 1 and u 2 before the interaction, and afterwards they are v 1 and v 2, then m1u1m2u2m1v1m2v2.displaystyle m_1u_1m_2u_2m_1v_1m_2v_2. This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and separations caused by explosive forces. 4 It can also be generalized to situations where newton's laws do not hold, for example in the theory of relativity and in electrodynamics. 6 Dependence on reference frame edit newton's apple in Einstein's elevator. In person A's frame of reference, the apple has non-zero velocity and momentum.

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2 3 Relation to dissertation force edit If the net force applied to a particle is a constant f, and is applied for a time interval Δ t, the momentum of the particle changes by an amount ΔpFΔt. Displaystyle delta pFDelta. In differential form, this is Newton's second law ; the rate of change of the momentum of a particle is equal to the instantaneous force f acting on it, 1 Fdpdt. If the net force experienced by a particle changes as a function of time, f(t), the change in momentum (or impulse j ) between times t 1 and t 2 is ΔpJt1t2F(t)dt. Displaystyle delta pJint _t_1t_2F(t. Impulse is measured in the derived units of the newton second (1 Ns 1 kgm/s) or dyne second (1 dynes 1 gm/s) Under the assumption of constant mass m, it is equivalent to write Fd(mv)dtmdvdtma, displaystyle Ffrac d(mv)dtmfrac dvdtma, hence the net force is equal to the. 1 Example : A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in. The net force required to produce this acceleration is 3 newtons due north. The change in momentum is 6 kgm/s.

In cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum resume is in gram centimeters per second (gcm/s). Being a vector, momentum has magnitude and direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kgm/s due north measured with reference to the ground. Many particles edit The momentum of a system of particles is the sum of their momenta. If two particles have respective masses m 1 and m 2, and velocities v 1 and v 2, the total momentum is pp1p2m1v1m2v2.displaystyle the momenta of more than two particles can be added more generally with the following: pimivi. A system of particles has a center of mass, a point determined by the weighted sum of their positions: r_textcmfrac m_1r_1m_2r_2cdots m_1m_2cdots frac sum limits _im_ir_isum limits _im_i. If all the particles are moving, the center of mass will generally be moving as well (unless the system is in pure rotation around it). If the center of mass is moving at velocity v cm, the momentum is: pmvcm. This is known as Euler's first law.

equation for deformable solids. Contents Newtonian edit momentum is a vector quantity : it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations (see multiple dimensions ). Single particle edit The momentum of a particle is conventionally represented by the letter. It is the product of two quantities, the particle's mass (represented by the letter m ) and its velocity ( v 1 pmv. The unit of momentum is the product of the units of mass and velocity. In si units, if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second (kgm/s).

If m is an object's mass and v is the velocity (also a vector then the momentum is pmv, displaystyle mathbf p mmathbf v, in, sI units, it is measured in kilogram meters per second ( kg m/s ). Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is also conserved in special relativity, (with a modified formula) and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity. It is an expression of one of the fundamental symmetries of space and time: translational symmetry. Advanced will formulations of classical mechanics, lagrangian and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems the conserved quantity is generalized momentum, and in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function. The momentum and position operators are related by the heisenberg uncertainty principle.

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