A particle moves on the x-axis as follows : it starts from rest at $t=0$ from the point $x=0$ and comes to rest at the point $x=1$ at $t=1$. No other information is available about its motion at intermediate time ($0<t<1$). If $\alpha$ denotes the instantaneous acceleration of the particle, then
A) $\alpha$ cannot remain positive for all t in the interval $0≤t≤1$B) |$\alpha$| cannot exceed $2$ at any point in its path
C) |$\alpha$| must be $≥4$ at some point or points in its path
D) $\alpha$ must change sign during the motion but no other assertion can be made with the information given.
I stumbled upon this kinematics question. However, I am more interested in the maths of it. I am sure that option A) is right because if acceleration was always positive, the particle would have never come to rest again.
However, I am not sure if any other piece of information can be extracted from this data.
This question is equivalent to the following mathematical statement:
"Given a real function $v(t)$ satisfying $$\int_{0}^{1}v(t) dt = 1$$ and $$v(0) = v(1) = 0,$$ what all can you conclude about the modulus of $\frac{dv}{dt}$?"
The other option that is also correct is option C). I cannot convince myself why should modulus of $\frac{dv}{dt}$ be $≥4$? I am not looking for intuition but for mathematical proofs?
What kind of calculus theorems can be applied here? I have tried using Rolle's theorem but that just confirmed that my choice option A) was right.
Can someone please help me out to extract more information from these given conditions? Any help will be appreciated. Thanks in advance.
