[Math] Velocity and acceleration problem

multivariable-calculus

  1. Show that if the dot product of the velocity and acceleration of a moving particle is positive (or negative), then the speed of the particle is increasing (or decreasing).

  2. If at all times t the position and velocity vectors of a moving particle satisfy $v(t)=r(t)$, and if $r(0)=r_0$, find $r(t)$ and the acceleration $a(t)$. What is the path of the motion?

Please help!

  1. Solution proposal

$u(t)^2$= v * v

I then differentiate both sides and get

$2u du/dt$=av+va=2va

$u du/dt$=va

Which IMO shows that when the dot product is positive so is the speed

Best Answer

for the first one notice that: $${\bf v}. {\bf a}={\bf v}.{d{\bf v}\over dt}={d\over dt}({1\over 2}{\bf v}. {\bf v})={d\over dt}({1\over 2}\|v\|^2)$$ for the second: $${d{\bf r}\over dt}={\bf r} \quad \Rightarrow \quad {\bf r}(t)={\bf r}(0)\, e^t={\bf r}_0\, e^t \\ \Rightarrow {\bf v}(t)={\bf a}(t)={\bf r}_0\, e^t$$ It is easy to see the path of the motion is a semi-line contains $\bf r_0$.

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