Question

what does the slope in displacement time^2 graph indicate? Explain with reasons.....
i want one answer of abhi178

Answer 1

There are many cases possible here .
but for understanding we use here three cases .

Let y - axis represent displacement and x - axis represent time² .

case 1:- when , displacement - time² graph is a straight line parallel to x- axis .then,
it means displacement is constant , independent of value of time² .
e.g displacement = K
but we know,
slope of graph = displacement /time²
= (displacement/time)/time
= velocity/time
=acceleration
here , graph parallel to x - axis ,
means , slope = 0
so, acceleration = 0

Case2 :- when, displacement - time² graph is a straight line but varies linearly.
it means ,
displacement directly proportional to time² .
e.g displacement = Kt²
where K is constant .
hence if we plot graph of displacement - time graph then, we will find parabolic graph. but displacement - time² graph , it's linear graph .
slope of graph = ( x2 - x1)/(t2² - t1²) = acceleration.
also we can find , acceleration by using differentiation rule .
x = Kt²
differentiate wrt t
dx/dt = 2Kt
hence, velpcity , directly proportional to time.

again, differentiate wrt t
d²x/dt² = 2K
hence, acceleration will be constant .

Case 3 :- when, displacement - time² graph is parabola then,
displacement (x) = K(t²)² = Kt⁴
differentiate wrt time,
dx/dt = 4Kt³
hence, velocity directly proportional to cube of time .

again, differentiate wrt time,
d²x/dt² = 12Kt²
hence, acceleration directly proportional to square of time .

Answer 2

Answer:

A sloping line on a displacement-time graph shows that the object is moving. In a displacement-time graph, the slope or gradient of the line, is equal to the velocity of the object. The steeper the line (and the greater the gradient) the faster the object is moving.