Chapter 3 - 3.5 - Mathematical Modeling and Variation - 3.5 Exercises - Page 296: 52

$K$ varies jointly as $m$ and the square of $v$

It means that if we multiply $m$ by a factor $a$ and $v$ by a factor $b$, then $K$ will be multiplied by a factor $ab^2$. $K_1=\frac{1}{2}m_1v^2_1$ Now suppose that $m_2=am_1$ and $v_2=bv_1$ $A_2=\frac{1}{2}m_2v^2_2=\frac{1}{2}am_1(bv_1)^2=\frac{1}{2}am_1b^2v_1^2=ab^2\frac{1}{2}m_1v_1^2=ab^2A_1$ $A_2 =ab^2A_1$