can 2 dimensional matter have velocity ?
Answers
Answer:
Average velocity
Consider a particle moving along a curved path in x-y plane shown below in the figure
Suppose at any time,particle is at the point P and after some time 't' is at point Q where points P and Q represents the position of particle at two different points.
Displacement vector in xy coordinate system
Position of particle at point P is described by the Position vector r from origin O to P given by
r=xi+yj
where x and y are components of r along x and y axis
As particle moves from P to Q,its displacement would be would be Δr which is equal to the difference in position vectors r and r'.Thus
Δr = r'-r = (x'i+y'j)-(xi+yj) = (x'-x)i+(y'-y)j = Δxi+Δyj (1)
where Δx=(x'-x) and Δy=(y'-y)
If Δt is the time interval during which the particle moves from point P to Q along the curved path then average velocity(vavg) of particle is the ratio of displacement and corresponding time interval
average velocity equation in two-dimension
since vavg=Δr/Δt , the direction of average velocity is same as that of Δr
Magnitude of Δr is always the straight line distance from P to Q regardless of any shape of actual path taken by the particle.
Hence average velocity of particle from point P to Q in time interval Δt would be same for any path taken by the particle.
3.Instantaneous velocity
We already know that instantaneous velocity is the velocity of the particle at any instant of time or at any point of its path.
If we bring point Q more and more closer to point P and then calculate average velocity over such a short displacement and time interval then
Instantaneous velocity in two dimensions
where v is known as the instantaneous velocity of the particle.
Thus, instantaneous velocity is the limiting value of average velocity as the time interval approaches zero.
As the point Q approaches P, direction of vector Δr changes and approaches to the direction of the tangent to the path at point P. So instantaneous vector at any point is tangent to the path at that point.
Figure below shows the direction of instantaneous velocity at point P.
graph showing the path of Instantaneous velocity in two dimensions
Thus, direction of instantaneous velocity v at any point is always tangent to the path of particle at that point.
Like average velocity we can also express instantaneous velocity in component form
Instantaneous velocity in two dimensions
where vx and vy are x and y components of instantaneous velocity.
Magnitude of instantaneous velocity is
|v|=√[(vx)2+(vy)2]
and angle θ which velocity vector makes with x-axis is
tanθ=vx/vy
Expression for instantaneous velocity is
Expression for Instantaneous velocity in two dimensions
Thus, if expression for the co-ordinates x and y are known as function of time then we can use equations derived above to find x and y components of velocity.
4. Average and instantaneous acceleration
Suppose a particle moves from point P to point Q in x-y plane as shown below in the figure
Average accleration vector representation in two dimensions
Suppose v1 is the velocity of the particle at point P and v2 is the velocity of particle at point Q
Average acceleration is the change in velocity of particle from v1 to v2 in time interval Δt as particle moves from point P to Q. Thus average acceleration is
Average accleration in two dimensions
Average acceleration is the vector quantity having direction same as that of Δv.
Again if point Q approaches point P, then limiting value of average acceleration as time approaches zero defines instantaneous acceleration or simply the acceleration of particle at that point. This, instantaneous acceleration is
instantaneous acceleration in two dimensions
Figure below shows instantaneous acceleration a at point P.
instantaneous acceleration at a point
Instantaneous acceleration does not have same direction as that of velocity vector instead it must lie on the concave side of the curved surface.
Thus velocity and acceleration vectors may have any angle between 0 to 180 degree between them.
Concept Map's for Velocity and acceleration in two dimensional motion
Concept Map for Velocity and acceleration in two dimensional motion
Question
The position of a object is given by
r= 3ti + 2t2j+ 11k
Where t is in second and coefficients have the proper units for r to be in centimetres
Answer:
Speed and velocity are both measured using the same units. The SI unit of distance and displacement is the meter. The SI unit of time is the second. The SI unit of speed and velocity is the ratio of two — the meter per second .