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A proton and a deuteron are initially at rest and are accelerated through the same potential difference. Which of the following is false concerning the final properties of the two particles ?
They have different speeds
They have same momentum
They have same kinetic energy
They have been subjected to same force
Solution
A deuteron is an isotope of hydrogen, having one proton and one neutron.
Proton and deuteron have the same charge. Let the charge be denoted by p.
Mass of proton is 1 amu, while mass of deuteron is 2 amu.
$m_{p}=1$ amu
$m_{d}=2 a m u$
Let they be accelerated by potential difference of $V$.
Kinetic energy of proton
$K_{p}=0.5 m_{p} v_{p}^{2}=e V$
Kinetic energy of deuteron
$K_{d}=0.5 m_{d} v_{d}^{2}=e V$
As we can see, $K_{p}=K_{d}$
$0.5 m_{d} v_{d}^{2}=0.5 m_{p} v_{p}^{2}$
$m_{d} v_{d}^{2}=m_{p} v_{p}^{2}$
Since $m_{d}=2 m_{p}$
$2 v_{d}^{2}=v_{p}^{2}$
$v_{p}=\sqrt{2} v_{d}$
So, velocity of proton is $\sqrt{2}$ times the velocity of deuteron.
Also.
$m_{p} v_{p}=0.5 m_{d} \times \sqrt{2} v_{d}=\frac{m_{d} v_{d}}{\sqrt{2}}$
Initial velocity of both the proton and deutron are zero. So, their initial momentum is zero.
Final momentum of proton and deutron are $m_{p} v_{p}$ and $m_{d} v_{d}$
So, we can find the change in momentum for both.
But we don't have any data to find the time taken for change in momentum.
since $F=\frac{d p}{d t}$ and we don't have dt
so we can't compare the force to which the two particles have been subjected.
