Answer to Question #104310 in Classical Mechanics for Samson Yoo

Question

1. A mass breaks suddenly into two parts, masses M and m, with speeds V and v respectively and carrying a total kinetic energy K. If M / m = R =, what fractions of K does each of the masses carry? (Think about significant figures.)

a) M carries _____ K.

b) m carries _____ K.

2. A machine gun fires bullets, each with mass m = 55 g at a speed of u = 450 m.s^-1. The gun fires 5.0 bullets per second. Let's use it as a rocket engine. What thrust does it produce?

3. Let's combine momentum and collisions with some of the other things we've learned. A small block of mass M = 0.201 kg hangs on the end of a light, in extensible string, length R = 25 cm. A small dart (of mass m=0.10 kg) traveling in the horizontal direction collides with and remains fixed in the block. What is the minimum speed v of the dart such that the combined object completes a circular path around the support point of the block? How fast must the block be traveling at the top?

Answer 1

1) From the conservation of momentum:

$$mv=MV=RmV\to v=RV$$

$$K=0.5mv^2+0.5Rm\frac{v^2}{R^2}=0.5mv^2\left(1+\frac{1}{R}\right)$$

a)

$$K_m=\frac{1}{\left(1+\frac{1}{R}\right)}K=\frac{R}{R+1}K$$

b)

$$K_M=\left(1-\frac{R}{R+1}\right)K=\frac{1}{R+1}K$$

2)

$$F=mu\frac{N}{t}$$

$$F=(0.055)(450)(5)=120\ N$$

3) From the conservation of momentum:

$$mv=(m+M)u$$

At the top:

$$m\frac{u_t^2}{R}=mg$$

$$u_t=\sqrt{gR}=\sqrt{(9.8)(0.25)}=1.57\frac{m}{s}$$

From the conservation of energy:

$$u^2=u_t^2+2Rg$$

$$u=\sqrt{3gR}$$

So,

$$mv=(m+M)\sqrt{3gR}$$

$$0.1v=(0.1+0.201)\sqrt{3(9.8)(0.25)}$$

$$v=8.2\frac{m}{s}$$