Question

A single conservative force $F\left(x\right)$ acts on $1\text{kg}$ particle that moves along $x$ axis. The potential energy $U\left(x\right)$ is given by $U\left(x\right)=10+(x-1{)}^2$. At $x=2\text{m}$, The particle has kinetic energy of $20\text{J}$. The maximum kinetic energy of the particle is

• A. $31\text{J}$
• B. $10\text{J}$
• C. $21\text{J}$
• D. $15\text{J}$

Answer 1

The correct option is C $21\text{J}$

According to law of conservation of energy, if no dissipative forces are present then the total mechanical energy remains conserved.
Here, sum of kinetic energy and potential energy remains conserved.
Potential energy at $x=2\text{m}$, $U_{x=2}=10+(2-1{)}^2$
$=10+1=11\text{J}$
Total mechanical energy $\left(E\right)$ at $x=2\text{m},$
$E=U_{x=2}+K{E}_{x=2}$
$E=11+20=31\text{J}$
$\because$ Total mechanical energy $\left(E\right)$ remains constant
Kinetic energy will be maximum when particle have minimum potential energy.
Minimum potential energy $U_{min}$ will be at $x=1\text{m},$
$U_{min}$$=10\text{J}$
So, maximum kinetic energy $=E-U_{\text{min}}$
$=31-10=21\text{J}$