Answer
$a,b,c,f$
Work Step by Step
To prove which equations the units are consistent, we need to substitute each variable by its units and simplify.
a) $x(m)=v(m/s)*t(s)$
$m=\frac{m}{s}*s$
$m=\frac{m*s}{s}$
$m=m$
This one is consistent
b) $x(m)=v(m/s)*t(s)+\frac{1}{2}*a(m/s^2)*(t(s))^2$
$m=\frac{m}{s}*s+\frac{1}{2}*\frac{m}{s^2}*s^2$
$m=\frac{m*s}{s}+\frac{1}{2}*\frac{m*s^2}{s^2}$
$m=m+\frac{1}{2}m=\frac{3}{2}m$
This one is consistent
c) $v(m/s)=a(m/s^2)*t(s)$
$\frac{m}{s}=\frac{m}{s^2}*s$
$\frac{m}{s}=\frac{m*s}{s^2}$
$\frac{m}{s}=\frac{m}{s}$
This one is consistent
d) $v(m/s)=a(m/s^2)*t(s)+\frac{1}{2}*a(m/s^2)*(t(s))^3$
$\frac{m}{s}=\frac{m}{s^2}*s+\frac{1}{2}*\frac{m}{s^2}*s^3$
$\frac{m}{s}=\frac{m*s}{s^2}+\frac{1}{2}*\frac{m*s^3}{s^2}$
$\frac{m}{s}=\frac{m}{s}+\frac{1}{2}ms$
This one is not consistent
e) $(v(m/s))^3=2*a(m/s^2)*(x(m))^2$
$\frac{m^3}{s^3}=2*\frac{m}{s^2}*m^2$
$\frac{m^3}{s^3}=2*\frac{m*m^2}{s^2}$
$\frac{m^3}{s^3}=2\frac{m^3}{s^2}$
This one is not consistent
f) $t(s)=\sqrt \frac{2*x(m)}{a(m/s^2)}$
$s=\sqrt {2*m*\frac{s^2}{m}}$
$s=\sqrt {2*\frac{m*s^2}{m}}$
$s=\sqrt {2*s^2}$
$s=\sqrt 2s$
This one is consistent
