Wondering from the position vs time graph of an object moving with constant acceleration. How could you find the velocity? So the position vs time graph would be a parabola. I am thinking that the instantaneous velocity, $v_y$, may be found as the slope of the tangent to a displacement versus time graph at any point. Wondering if someone could help clarify this for me
The instantaneous velocity at a point in time is equal to the average velocity over any time interval that the point is in the middle of. So if you want the instantaneous velocity at 5 seconds, find the slope of the line between points on the curve at 4 and 6 seconds, or 3 and 7 seconds, etc.
Note: Someone apparently downvoted the above answer without understanding, so I must have been too brief or unclear. I'll explain. The secant line to a continuous curve is defined as the line connecting any two points on the curve. In the interval between those two points for an increasing function, it can be seen that the slope of the tangent to any point changes from a value less than that of the secant line to a value greater than that of the secant line. So there must be a point where the slope of the tangent is the same as that of the secant line (in other words, they are parallel). This is the idea that leads to the derivative of a function in calculus. For a parabolic position vs. time curve, that point is located in the middle of the time interval. The slope of the tangent at that point, and the slope of the secant line that it's in the middle of, are equal to the instantaneous velocity.
For unidirectional uniform motion,average velocity,average speed,instantaneous velocity and instantaneous speed all are equal.
Things are not so complicated even if we are dealing with accelerated motion.Just find the point at which you want the instantaneous velocity and calculate its slope.it will give you instantaneous velocity.
Yes, finding velocity that way is correct.
Acceleration is constant $=a$.
Once integrate, velocity is linear
$$ v = v_0 + a\,t $$
Again integrate, position is parabolic
$$ x = x_0+ v_0 \,t + a t^2/2. $$
Displacement is a product of velocity and total time of travel.
If you see a velocity time graph (please see the explanation i've written in page below), you’ll see that it is composed of several small rectangles under the curve that are nothing but products of small time intervals (often shown as “dt”) and the velocity at that time. If you sum up the area of these rectangles or integrate (integration is nothing but summation of area), you get the total displacement
You can also watch this video made by me for more clarity-