Average velocity during a particular time interval for a particle is a vector quantity whose x-component is the change in x divided by the time interval.
The x-component of average velocity, or average x-velocity, is the x component of displacement,Δx, divided by the time interval,Δt, during which the displacement occurs.
The rule for the sign of velocity depends on four main case scenarios:
The velocity at a specific instant of time or specific point along the path is known as instantaneous velocity.
==In the language of calculus, the limit of Δx/Δt as Δt approaches zero is called the derivative of x with respect to t and is written dx/dt.==
On a graph of position as a function of time for straight line motion, the instantaneous x-velocity at any point is equal to the slope of the tangent to the curve at that point.
A motion diagram shows the particle’s position at various instants as well as arrows to represent the particle’s velocity at each instant.
The change in the x-component of velocity Δv(x), divided by the time interval Δt is known as average acceleration.
The instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero. In the language of calculus, instantaneous acceleration equals the derivative of velocity with time.
The rules for the sign of the acceleration are:
On a graph of x-velocity as a function of time, ==the instantaneous x-acceleration at any point is equal to the slope of the tangent to the curve at that point==.
Tangents drawn at different points along the curve have different slopes, so the instantaneous x-acceleration varies with time.
The second derivative of any function is directly related to the concavity or curvature of the graph of that function.
Here, the acceleration is the double derivative of dx/dt.
Where the x-t graph is concave up (curved upward), the x-acceleration is positive and is increasing.
At a point where the x-t graph is concave down (curved downward), the x-acceleration is negative and is decreasing.
At a point where the x-t graph has no curvature, such as an inflection point, the x-acceleration is zero and the velocity is not changing.
The three equations for motion in a straight line and in constant acceleration are:
Here the vx is the final velocity, v0x is the initial velocity, ax is the acceleration, t is the time, x is the final position and x0 is the starting position.
Free fall is a case of motion with constant acceleration.
When the acceleration is variable(not constant), ==we have to integrate the acceleration at each point in the distance==. Thus the equations become:
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