Step 1: Displacement equations
The proton and the alpha particle both start from rest, so their displacement after some time is given by:
s=‌at2.Since they travel the same distance:
‌aαtα2=‌aptp2This means:
aαtα2=aptp2Step 2: Acceleration in an electric field
The acceleration of a charged particle in an electric field
E is:
a=‌- For a proton, charge
=e, mass
=mp- For an alpha particle, charge
=2e, mass
=4mpSo:
−ap=‌−aα=‌=‌Step 3: Substitute accelerations into equation
Plug the accelerations into the earlier equation:
(‌)tα2=(‌)tp2Step 4: Solve for time ratio
Divide both sides by
‌ :
‌=tp2This gives:
tα2=2tp2Taking square roots:
tα=√2tpSo,
‌=‌or
tp:tα=1:√2
