Question

My question is from the textbook University Physics with Modern Physics 14th edition. Authors: Young and Freedman. It is from Chapter 23 Problem 52. The question is:

A proton and an alpha particle are released from rest when they are 0.225 nm apart. The alpha particle (a helium nucleus) has essentially four times the mass and two times the charge of a proton. Find the maximum speed and maximum acceleration of each of these particles. When do these maxima occur: just following the release of the particles or after a very long time?

Thank you!

Answer 1

a)

proton mass, m1=1.67*10^-27 kg

alpa particle mass, m2=4*m1

separation, d=0.225 nm

by using law of conservation momentum,

m1*u1+m2*v2=m1*v1+m2*v2

0=m1*v1+m2*v2

0=m1*v1+4*m1*v2

0=(v1+4*v2)*m1

0=v1+4v2 ---(1)

by using law of conservation of energy,

k*q1*q2/d=1/2*m1*v1^2 + 1/2*m2*v2^2

k*e*2e*/d=1/2*m1*v1^2 + 1/2*4*m1*v2^2

k*2e^2*/d=1/2*m1*(v1^2 + 4*v2^2)

9*10^9*2*(1.6*10^-19)^2/(0.225*10^-9)=1/2*1.67*10^-27*(v1^2+4*v2^2)

2.048*10^-18=1/2*1.67*10^-27*(v1^2+4*v2^2) ----(2)

from equation no (1) and (2)

speed of proton, v1=44296.23 m/sec

speed of alpha particel, v2=11074.06 m/sec

b)

F=K*q1*q2/r^2

F=9*10^9*e*2e/r^2

F=9*10^9*2*(1.6*10^-19)^2/(0.225*10^-9)^2

F=9.1*10^-9 N

m1*a1=9.1*10^-9

1.67*10^-27*a1=9.1*10^-9

====> a1=5.45 *10^18 m/sec^2

acceleration of proton, a1=5.45*10^18 m/sec^2

and

F=9.1*10^-9 N

m2*a2=9.1*10^-9

4*1.67*10^-27*a2=9.1*10^-9

====> a2=1.36 *10^18 m/sec^2

acceleration of alpha particle, a2=1.36*10^18 m/sec^2

c)

these speed and acceleration wiill be maximum at initially when realsed