Q.What physical quantity is represented by

[!TLDR] Slope of an x-t graph = instantaneous velocity (dx/dt); slope of a v-t graph = instantaneous acceleration (dv/dt). [!ANSWER] x …

Slope of a position-time (x-t) graph. By definition, instantaneous velocity is $v = dx/dt$ — the derivative of position with respect to time (Section 2.4). Geometrically, the derivative at a point on a graph is the slope of the tangent to the curve at that point. So the slope of an x-t graph, at any instant, IS the instantaneous velocity at that instant. A horizontal x-t graph (zero slope) means the body is at rest; a steep positive slope means a large positive velocity; a negative slope means the body is moving in the negative direction.

Slope of a velocity-time (v-t) graph. By an exactly analogous argument, instantaneous acceleration is $a = dv/dt$ (Section 2.5) — the derivative of velocity with respect to time. So the slope of a v-t graph, at any instant, is the instantaneous acceleration at that instant. A horizontal v-t graph (zero slope) means zero acceleration (uniform velocity); a straight, uniformly sloped v-t graph means constant (uniform) acceleration, as developed fully in Sections 2.7 and 2.8. …

Apply the calculus definitions v = dx/dt and a = dv/dt directly, interpreting each derivative geometric …

• Reversing the two: saying the x-t slope gives acceleration or the v-t slope gives velocity
• Forgetting that a non-straight x-t or v-t graph still gives INSTANTANEOUS values via a tangent, not a single overall slope …