Given information:
- A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm.
- The bullet faces constant resistance to motion.

To determine:
- How much further the bullet will penetrate before coming to rest.

Solution:
Let's assume the initial velocity of the bullet is V.

Step 1: Calculate the velocity after penetrating 3 cm
Since the bullet loses half of its velocity after penetrating 3 cm, the velocity after penetrating 3 cm can be calculated using the formula:

V_after = V_initial / 2

Step 2: Calculate the time taken to penetrate 3 cm
We can use the equation of motion to calculate the time taken to penetrate 3 cm. The equation is:

s = ut + (1/2)at^2

where:
- s is the distance traveled (3 cm or 0.03 m)
- u is the initial velocity (V)
- a is the acceleration (constant resistance to motion, which can be denoted as -k, where k is a positive constant)
- t is the time taken to travel the distance

Plugging in the given values, the equation becomes:

0.03 = V * t + (1/2)(-k)t^2

Step 3: Calculate the time taken to come to rest
When the bullet comes to rest, its final velocity is 0. Therefore, we can use the equation of motion to calculate the time taken to come to rest. The equation is:

v = u + at

where:
- v is the final velocity (0)
- u is the initial velocity (V_after)
- a is the acceleration (constant resistance to motion, which can be denoted as -k)
- t is the time taken to come to rest

Plugging in the given values, the equation becomes:

0 = V_after + (-k)t_rest

Step 4: Calculate the distance traveled after penetrating 3 cm
To find the distance traveled after penetrating 3 cm, we can use the equation of motion:

s = ut + (1/2)at^2

where:
- s is the distance traveled after penetrating 3 cm (let's denote it as S)
- u is the initial velocity after penetrating 3 cm (V_after)
- a is the acceleration (constant resistance to motion, which can be denoted as -k)
- t is the time taken to come to rest (t_rest)

Plugging in the given values, the equation becomes:

S = V_after * t_rest + (1/2)(-k)t_rest^2

Step 5: Substitute the values and calculate the distance
We have two equations from Step 2 and Step 4, which can be solved simultaneously to find the values of t and S.

From Step 2: 0.03 = V * t + (1/2)(-k)t^2
From Step 4: S = V_after * t_rest + (1/2)(-k)t_rest^2

Substituting V_after = V_initial / 2 and rearranging the equations, we get:

0.03 = V * t + (-k/2)t^2
S = (V_initial / 2) * t_rest +