Momentum Conservation and Center of Mass

[TwinGemini14]

1. Ball 1, ball 2, and ball 3 have masses 1 kg, 2 kg, and 3 kg respectively, and are initially arranged at x = 5 m, x = 10 m, and x = 15 m respectively (see picture). y = 5 m for all three balls. Now suppose that a net force of 1 N is applied to ball 1 in the +y direction and a net force of 2 N is applied to ball 2 in the -y direction. The net force on ball 3 is zero.


What is the magnitude of the acceleration of the center of mass of the three-ball system?

A) 0.00 m/s2
B) 0.17 m/s2
C) 0.33 m/s2
D) 1.00 m/s2
E) 1.33 m/s2

----

So I first drew out the picture of the masses. Then since the acceleration of the center of mass is just the weighted average acceleration of all the masses, I did...
F1 = m1a1 or 1 = 1(a1) so a1=1.
F2 = m2a2 or 2 = 2(a2) so a2=1.
F3 = m3a3 or 0 = 3(a3) so a3=0.

(a1+a2+a3)/(m1+m2+m3) = 0.333m/s^2. So the answer is C. Is this correct?

============
A block of mass 50 kg is sliding along in the +x direction (to the right) with initial magnitude of momentum p0 = 2450 kg*m/s on a horizontal frictionless surface.


Then a constant force F = 14 N is applied as shown in the above drawing, where the angle of the force above horizontal is 20°.

2) One second after the force is applied, the momentum of the block in the ground frame will have:

A) decreased in magnitude
B) been conserved
C) increased in magnitude

----
The momentum has increased in magnitude since the object is accelerating due to a net force, correct? C?
----
3) After the force is applied, the total mechanical energy of the block will:

A) be less than it was before the force was applied
B) remain the same as it was before the force was applied
C) be greater than it was before the force was applied

----
B. Since the total mechanical energy is always conserved in this situation, right?
----

Can somebody please look over my work to see if I did these right? I really appreciate it. Thank you!

 

[LowlyPion]

(a1+a2+a3)/(m1+m2+m3) = 0.333m/s^2. So the answer is C. Is this correct?

No. Why do you think the sum of the accelerations divided by total mass is somehow still an acceleration? Units are wrong at a minimum.

Also too your forces are in different directions. Maybe if you considered the Net force divided by total mass you would have a better handle on how the CoM is being accelerated?

 

[LowlyPion]

For 2) assuming your force is directed horizontally in the direction of movement already, then yes momentum would increase with velocity.

For the next part, what is the definition of mechanical energy?