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Equation [3] involves the average velocity v + v 0 / 2 . Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from v 0 to v, as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows ...
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.)
In terms of a displacement-time (x vs. t) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point, and the average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity.
For rod length 6" and crank radius 2" (as shown in the example graph below), numerically solving the acceleration zero-crossings finds the velocity maxima/minima to be at crank angles of ±73.17615°.
A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function , which is defined by the formula: [ 1 ]
A rocket's required mass ratio as a function of effective exhaust velocity ratio. The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the ...
With this form for the displacement, the velocity now is found. The time derivative of the displacement vector is the velocity vector. In general, the derivative of a vector is a vector made up of components each of which is the derivative of the corresponding component of the original vector. Thus, in this case, the velocity vector is:
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.