A 5 g bullet passes through a 1 kg wooden sign that was hanging at rest on a 0.2 m long rope. After passing through, the bullet loses half of its velocity, and the sign gains just enough velocity to swing around in a complete vertical circle. Assume that the rope is strong enough so that it won’t break, and assume that it doesn’t get any shorter as it swings around. What was the initial velocity of the bullet?

m1=0.005 kg, m2= 1 kg, R=0.2 m.

v=?

Law of conservation of momentum
m1•v = m1•(v/2) +m2•u
m1•v/2= m2•u,
u= m1•v/2•m2. (1)
Law of conservation of energy
KE =PE +KE1
m2•u²/2=m2•g•2R+m2•u1²/2
u²/2= g•2R+ u1²/2 . (2)

At the highest point
m2•a=m2•g
m2•u1²/R=m2•g.
u1=sqrt(gR)=
sqrt(9.8•0.2)=1.4 m/s. (3)

Substitute (1) in (2)
(m1•v) ² /(2•m2)² 2= g•2R+ u1²/2 .
v=(2•m2/m1)sqrt(4gR+u1²)=
=(2•1/0.005)sqrt(4•9.8•0.2+1.4²)=
=400•3.13=1252 m/s.
Check my calculations.

it seems right, but the answer is 1120m/s?

Your answer (1120 m/s)is for the case when the block ( sign) is suspended by stiff rod of length R and of negligible mass. But your given data is "the rope"

To find the initial velocity of the bullet, we can start by analyzing the conservation of momentum and the conservation of energy in this scenario.

First, let's consider the conservation of momentum. The initial momentum of the bullet before passing through the wooden sign is given by:

Initial momentum of bullet = mass of bullet x initial velocity of bullet

Momentum is conserved in this scenario since there are no external forces acting on the system. Therefore, the final momentum of the system after the bullet passes through the sign is equal to the initial momentum. The final momentum is given by:

Final momentum of system = (mass of bullet + mass of sign) x final velocity of sign

Since the sign was initially at rest, the final velocity of the sign is the velocity required to swing around in a complete vertical circle.

Now, let's consider the conservation of energy. Initially, both the bullet and the sign have kinetic energy due to their motion. After the bullet passes through the sign, some of the bullet's initial kinetic energy is transferred to the sign, causing it to gain velocity and swing around. This means that the bullet loses half of its velocity and, therefore, half of its kinetic energy. Meanwhile, the sign gains enough velocity to move in a vertical circle.

The equation for the conservation of energy in this scenario is:

Initial kinetic energy of bullet = final kinetic energy of bullet + final kinetic energy of sign

Since the bullet loses half of its velocity, its final kinetic energy is half of its initial kinetic energy. The final kinetic energy of the sign can be calculated using the formula for the kinetic energy of an object in circular motion.

Now that we have expressions for both momentum and energy conservation, we can set them equal to each other and solve for the initial velocity of the bullet.

Let's denote the mass of the bullet as "m" and the mass of the sign as "M." We can substitute the given values:

m = 5 g = 0.005 kg (converting grams to kilograms)
M = 1 kg

From the problem statement, the bullet loses half of its velocity, so the final velocity of the bullet is half of its initial velocity:

final velocity of bullet = 0.5 x initial velocity of bullet

Now, we also know that the sign gains enough velocity to swing around in a complete vertical circle. This means that it has sufficient velocity to counteract the gravitational force acting on it throughout the circular motion.

The required velocity for circular motion can be calculated using the following formula:

v = sqrt(g x r)

Where "g" is the acceleration due to gravity and "r" is the length of the rope.

Given:
g = 9.8 m/s^2
r = 0.2 m

Plug in these values to find the required velocity for circular motion.

Now, by setting the expressions for momentum and energy conservation equal to each other and solving for the initial velocity of the bullet, we can find the answer to the question.

Final momentum of system = Initial momentum of bullet

(m + M) x final velocity of sign = m x final velocity of bullet

Substituting the values we have:

(0.005 kg + 1 kg) x final velocity of sign = 0.005 kg x 0.5 x initial velocity of bullet

Simplify the equation and solve for the final velocity of the sign:

final velocity of sign = 0.005 kg x 0.5 x initial velocity of bullet / (0.005 kg + 1 kg)

Finally, substitute the known values into the equation for the required velocity for circular motion and solve for the initial velocity of the bullet:

sqrt(g x r) = final velocity of sign

Now, you can calculate the initial velocity of the bullet by solving the equation.