In a common physics demonstration, two identical carts having rigid metal surfac
ID: 1410925 • Letter: I
Question
In a common physics demonstration, two identical carts having rigid metal surfaces and equal speeds collide with each other. Each cart has a piece of Velcro® at one end and a spring at the other end. See (Figure 1).
Part A
For collision (a) shown in the figure, find the magnitude of the velocity of the left cart after the collision.
Express your answer in terms of the initial velocity v0.
Part C
For collision (a) shown in the figure, find the magnitude of the velocity of the right cart after the collision.
Express your answer in terms of the initial velocity v0.
Part E
For collision (b) shown in the figure, find the magnitude of the velocity of the left cart after the collision.
Express your answer in terms of the initial velocity v0.
Explanation / Answer
Let us consider an elastic one dimensional collision b/w two particles of mass m1 and m2 suppose initial velocity U1 and u2
Applying law of conservation of linear momentum
We have
m1u1+m2u2=m1v1+m2v2
m1(u1-v1)=m2(v2-u2).....................(1)
According to law of conservation of kinetic energy
We have
1/2 m1u1^2+1/2m2u2^2=1/2m1v1^2+1/2m2v2^2
m1(u1^2-v1^2)= m2(v2^2-u2^2).............(2)
Dividing eq2 &eq1 we get
u1+v1=v2+u2
(u1-u2)=-(v1-v2)..............(3)
From eq3 v1=v2+u2-u1
Substituting this value of v1 in eq1 we get
m1[u1-(v2+u2-u1)]=m2(v2-u2)
2m1u1-m1v2-m1u2=m2v2-m2u2
2m1u1+(m2-m1)u2=(m1+m2)v2
=>v2=[2m1/m1+m2]u1+[m2-m1/m1+m2]u2........(4)
Further from eq3 v2=u1+v1-u2
Substituting this value v2 in eq1 we get
m1(u2-v1)=m2[(u1+v1-u2)-u2]
m1u1-m2u1+2m2u2=m2v1+m1v1
Therefore
V1=[2m2/m1+m2]u2+[m1-m2/m1+m2]u1.............(5)
Now we use above condition eq4&eq5
(a) ans
m1=m2=m
u1=u2=u
The velocity of left cart after collision
V1=[m-m/m+m]vo+[2m/m+m]vo
V1=go
After collision also they travel same velocity
(C) ans
The velocity of right cart after collision
V2=2mvo/m+m +[m-m]/2mvo
V2=go
The right cart also travel same velocity after collision
(E) ans
The velocity of left cart after collision is also travelled same velocity
V1=vo