Practice Problems with Springs
60. Two carts of equal mass m = 0.250 kg are placed on a frictionless track that has a light spring of force constant k = 50.0
N/m attached to one end of it, as in Figure P6.60. The red cart is given an initial velocity of v0 = 3.00 m/s to the right, and the
blue cart is initially at rest. If the carts collide elastically, find (a) the velocity of the carts just after the first collision and (b)
the maximum compression of the spring.
FIGURE P6.60
6.60
(a) Let m be the mass of each cart. Then, if v 0 is the initial velocity of the red cart, applying
conservation of momentum to the collision gives
m v b  m v r  m v 0  0 , or v b  v r  v 0
(1)
where v b and v r are the velocities of the blue and red carts after collision.
In a head-on elastic collision, we have v 2 f  v 2i  v1 f  v1i which reduces to
vb  vr  v0 .
Solving (1) and (2) simultaneously gives v r  0 , and vb  3.00 m s .
(b)
 KE  PEs  f
Using conservation of mechanical energy for the blue cart-spring system,
  KE  PEs i becomes
1
1
0  k x 2  mv b2  0
2
2
or
x
0.250 kg
m
vb 
 3.00 m s  0.212 m .
k
50.0 N m
(2)
61. A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large
spring, initially unstretched and with force constant k = 2.00 x 104 N/m, as in Figure P6.61. The cannon fires a 200-kg
projectile at a velocity of 125 m/s directed 45.0° above the horizontal. (a) If the mass of the cannon and its carriage is 5 000
kg, find the recoil speed of the cannon. (b) Determine the maximum extension of the spring. (c) Find the maximum force the
spring exerts on the carriage. (d) Consider the system consisting of the cannon, carriage, and shell. Why or why not?
FIGURE P6.61
6.61
(a) Use conservation of the component of
momentum in the horizontal direction
from just before to just after the cannon
firing.
 px  f   px i
gives
mshell  v shell cos 45.0  mcannon v recoil  0 , or
 m

v recoil    shell  v shell cos 45.0
 mcannon 
 200 kg 
 
125 m s  cos 45.0   3.54 m s
 5000 kg 
(b) Use conservation of mechanical energy for the cannon-spring system from right after the
cannon is fired to the instant when the cannon comes to rest.
 KE  PE
g
 PEs
   KE  PE
f
g
 PEs

i
1 2
1
2
0  0  kx max
 mcannon v recoil
00
2
2
x max
(c)
2
mcannon v recoil


k

 5000 kg  -3.54 m s 
2.00  10 4 N m
2
 1.77 m

Fmax  k x max  2.00  104 N m 1.77 m   3.54  104 N
(d) No. The rail exerts a vertical external force (the normal force) on the cannon and prevents it
from recoiling vertically. Momentum is not conserved in the vertical direction. The spring
does not have time to stretch during the cannon firing. Thus, no external horizontal force is
exerted on the system (cannon plus shell) from just before to just after firing. Momentum is
conserved in the horizontal direction during this interval.