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#1 In mathematics, the dot product or scalar product[note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called "the" inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
#2 The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction of motion (e.g. 60 km/h to the north). Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.
#3 Triangle law of vector addition is one of the vector addition laws. Vector addition is defined as the geometrical sum of two or more vectors as they do not follow regular laws of algebra. The resultant vector is known as the composition of a vector.
#4 In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement SI unit is Joule (J)
#5 •Scalars are quantities that are fully described by a magnitude (or numerical value) alone.
•Vectors are quantities that are fully described by both a magnitude and a direction.
#7 Law of conservation of linear momentum states that total momentum of the system is always conserved if no external force acts on an object or system of objects.
Consider a collision between two balls wherein there occurs no energy losses during the collision.
Momentum of the two balls before collision,
p1i = m1u1
p2i = m2u2
Total momentum of the system of two balls before collision, pi = p1i + p2i = m1u1 + m2u2.
During the collision m1 exerts an action force F12 on m2.
In response, from Newton"s third law, m2 exerts a reaction on m1, that is, F21, such that F12 = -F21
Negative sign implies that the two forces are directed in opposite directions.
After the collision they undergo change in velocity and the corresponding change in momentum. Momenta of the two balls after collision p1f = m1v1
p2f = m2v2
Total momentum of the system of two balls after collision, pf = p1f +p2f = m1v1 + m2v2
Also from Newton"s Second Law:
begin mathsize 12px style Force equals fraction numerator Change space in space momentum over denominator Time space interval end fraction
rightwards arrow straight F subscript 12 equals fraction numerator Change space in space momentum space produced space in space mass space straight m subscript 2 over denominator Collision space time end fraction
rightwards arrow straight F subscript 12 equals fraction numerator straight m subscript 2 straight v subscript 2 minus straight m subscript 2 straight u subscript 2 over denominator straight t end fraction
Similarly comma
rightwards arrow straight F subscript 21 equals fraction numerator straight m subscript 1 straight v subscript 1 minus straight m subscript 1 straight u subscript 1 over denominator straight t end fraction
From space Newton apostrophe straight s space third space law comma space straight F subscript 12 equals negative straight F subscript 21
rightwards arrow open parentheses fraction numerator straight m subscript 2 straight v subscript 2 minus straight m subscript 2 straight u subscript 2 over denominator straight t end fraction close parentheses equals negative open parentheses fraction numerator straight m subscript 1 straight v subscript 1 minus straight m subscript 1 straight u subscript 1 over denominator straight t end fraction close parentheses
rightwards arrow straight m subscript 1 straight u plus straight m subscript 2 straight u subscript 2 equals straight m subscript 1 straight v subscript 1 plus straight m subscript 2 straight v subscript 2 end style
Total momentum before collision = Total momentum after collision
Hence this equation implies that if no external force acts on the system of two colliding balls, the total momentum is conserved.
Explanation:
The tag answer is 6
