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Zach Forney
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What does v equals dx/dt mean? I interpret v as: the limiting value as a vanishingly small value for time t (dt) goes to 0. Or lim as dt-->0 of dx/dt.
Zach Forney said:What does v equals dx/dt mean? I interpret v as: the limiting value as a vanishingly small value for time t (dt) goes to 0. Or lim as dt-->0 of dx/dt.
Zach Forney said:What does v equals dx/dt mean?
Philip Wood said:Fine if v is the component of velocity in a given direction, but if v is velocity in its general, vector, sense, it has components in three dimensions, so it's hard to see what single graph it would be the slope of.
Absolutely not - this would imply that velocity is always directed exactly away from the origin.Drakkith said:Hmmm. Perhaps it would be the derivative of the magnitude of the displacement wrt time, which you would then multiply by the displacement's unit vector?
Philip Wood said:Fine if v is the component of velocity in a given direction, but if v is velocity in its general, vector, sense, it has components in three dimensions, so it's hard to see what single graph it would be the slope of.
I assumed Drakkith intended to say: plot distance along path against time, so slope of graph gives speed, then multiply by instantaneous unit vector in the direction the body is moving (that is by the magnitude of the velocity vector).MrAnchovy said:Absolutely not - this would imply that velocity is always directed exactly away from the origin.
dx/dt is a notation for x'(t), which by definition is equal to ##\lim_{\Delta t\to 0}\frac{\Delta x}{\Delta t}##. So you shouldn't put a "lim" in front of dx/dt.Zach Forney said:What does v equals dx/dt mean? I interpret v as: the limiting value as a vanishingly small value for time t (dt) goes to 0. Or lim as dt-->0 of dx/dt.
Ah yes, I think I misinterpreted what he was saying - sorry Drakkith!Philip Wood said:I assumed Drakkith intended to say: plot distance along path against time, so slope of graph gives speed, then multiply by instantaneous unit vector in the direction the body is moving (that is by the magnitude of the velocity vector).
This phrase refers to the mathematical relationship between velocity and distance. It means that velocity is the rate of change of distance over time, or in other words, how fast an object is moving in a particular direction.
Velocity and distance are related by the concept of differentiation in calculus. Velocity is the derivative of distance, meaning it is the slope of the distance-time graph.
Taking the derivative of distance involves finding the rate of change of distance with respect to time. This allows us to determine the velocity of an object at any given point in time.
The derivative of distance is important because it helps us understand the motion of objects. It allows us to calculate the velocity, acceleration, and other important parameters that describe an object's movement.
Imagine a car traveling at a constant speed of 60 miles per hour. After 1 hour, the car will have traveled a distance of 60 miles. The derivative of this distance is the car's velocity, which is 60 miles per hour. If the car's speed changes, the derivative of distance will also change accordingly.