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Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B. More complex trajectories require integrating the proper time between the respective events along the curve (i.e. the path integral ) to calculate the ...
at any two points in the flowing liquid. Here v {\displaystyle v} is fluid speed, g {\displaystyle g} is the acceleration due to gravity, y {\displaystyle y} is the height above some reference point, p {\displaystyle p} is the pressure, and ρ {\displaystyle \rho } is the density.
Minkowski diagrams are two-dimensional graphs that depict events as happening in a universe consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis.
Fick's first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low ...
These relationships can be demonstrated graphically. The gradient of a line on a displacement time graph represents the velocity. The gradient of the velocity time graph gives the acceleration while the area under the velocity time graph gives the displacement. The area under a graph of acceleration versus time is equal to the change in velocity.
Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of v = μmg , where m is the mass of the particle, g is the acceleration due to gravity, and μ is the particle's mobility in the fluid.
The principle asserts for N particles the virtual work, i.e. the work along a virtual displacement, δr k, is zero: [9] = (+) = The virtual displacements , δ r k , are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system at an instant of time , [ 22 ] i.e. in such a ...