# Acceleration-distance graphs

I solved the above quite theoretically, differentiating with respect to time the equation of the given graph $$v=mx+v_0,$$ where $$m=\frac{-v_0}{x_0}$$ is negative and $$v_0$$ is positive. Thus, $$a=m^2x+mv_0,$$ which means that acceleration as a function of distance when graphed should have a positive slope and a negative vertical intercept, rendering option A as correct.

But what I don't get is the intuition behind it. I've always solved graphs intuitively without equations and stuff and my first guess was C because the acceleration must be like constant? for the velocity distance graph to turn out to be a straight line? It's kinda my guess that if acceleration itself varied with distance then velocity graph would be curved (basically intuition, no hard proof). But then C has slope $$0$$ which disagrees with my intuition. If I'm moving with a constant retardation, my velocity will keep on decreasing linearly, won't it? Please help me understand this, without involving maths.

• Rew, it would be a good idea for you to develop more trust in following mathematical solutions rather than depending on "intuition". In my experience, there are many physics problems where the math takes unusual and non-intuitive twists and turns before turning into an answer that can be associated with a physical phenomenon. Commented May 9, 2020 at 17:57

The acceleration is not constant (it depends on x which changes with time). For a constant a, you would need v = vo + mt, (not mx).

If I'm moving with a constant retardation, my velocity will keep on decreasing linearly, won't it?

Yes, your velocity will indeed decrease linearly but with respect to time. In other words if the Velocity versus time graph is a straight line then your statement is correct but this is not true in case of velocity displacement graph.

The reason behind this is that: The slope of the velocity time graph gives acceleration hence if it is a constant then acceleration is a constant The slope of the velocity displacement graph gives the velocity gradient which, if constant, doesn't imply that the acceleration is constant

if acceleration itself varied with distance then velocity graph would be curved

This statement is not necessarily true. However it is true if acceleration varied with time.

In conclusion, intuition works well only when the graph is with respect to time but it's better to not use it in case of graphs drawn with displacement on the x axis.