A mass m1 = 7.65 kg, is at rest on a frictionless horizontal surface and connected to a wall by a spring with k = 67.9 N/m, as shown in the figure. A second mass, m2 = 5.29 kg, is moving to the right at v0 = 15.3 m/s. The two masses collide and stick together.

the figure is
htt p://s2.dosya.tc/server22/DIDqtx/P043figure.png. html

a) What is the maximum compression of the spring?
(in m) 2.731 m

b) How long will it take after the collision to reach this maximum compression?
(in s) 0.6857 s

i tried everything ,just found out the second one but still have a problem about the first one.

First find the speed after collision from conservation of momentum

initial momentum:
m v = 5.29 * 15.3 = 80.9 which will be momentum after crash

after crash
m = 5.29+7.65 = 12.94 kg
so for v after crash:
12.94 v = 80.9
v = 6.25 m/s after crash
v at end is zero and energy after crash is conserved so
(1/2)(12.94)(6.25)^2 = (1/2) (67.9) x^2
x^2 = 7.45
and
x = 2.73 meters I agree

time = 1/4 of period T
T = 2 pi sqrt(m/k)
= 2 pi sqrt (12.94/67.9)
= 2.74 seconds
1/4 T = .6857 seconds sure enough

thank you!

To find the maximum compression of the spring, we can use the principle of conservation of momentum. Before the collision, the first mass, m1, is at rest, so its initial momentum is zero. The second mass, m2, is moving to the right at a velocity of v0=15.3 m/s.

During the collision, the two masses collide and stick together, forming a combined mass (m1+m2). Since they stick together, their final velocity after the collision is the same.

We can use the equation for conservation of momentum to determine the final velocity:

(m1+m2) * vf = m2 * v0

Solving for vf, the final velocity after the collision:

vf = (m2 * v0) / (m1 + m2)

Now, to find the maximum compression of the spring, we need to consider the potential energy stored in the spring. The maximum compression occurs when all the kinetic energy is converted into potential energy of the spring.

The kinetic energy before the collision is given by:

KE = (1/2) * m2 * v0^2

The potential energy of the spring is given by:

PE = (1/2) * k * x^2

where k is the spring constant and x is the compression of the spring.

Since the sum of the kinetic and potential energies is conserved, we have:

KE + PE = (1/2) * (m1 + m2) * vf^2

Substituting the values and solving for x, the maximum compression:

(1/2) * m2 * v0^2 + (1/2) * k * x^2 = (1/2) * (m1 + m2) * vf^2

Plugging in the given values, we can solve for x to find the maximum compression of the spring.