A 14g bullet is traveling at 306m/s when it strikes a block of wood. If the block of wood exerts a force of 50,000 N opposing the motion of the bullet, how far will the bullet penetrate the block of wood?
To find how far the bullet will penetrate the block of wood, we can use the concept of work. When an object is subjected to a force and moves a certain distance, work is done. The work done on the bullet will be equal to the work done by the opposing force applied by the block of wood.
Work (W) is given by the formula:
W = Force × Distance × cos(θ)
In this case, the force exerted by the block of wood is opposing the motion of the bullet, so the angle θ between the force and the displacement will be 180 degrees.
Given:
Mass of the bullet (m) = 14 g = 0.014 kg
Initial velocity of the bullet (u) = 306 m/s
Force exerted by the block of wood (F) = 50,000 N
First, let's calculate the deceleration of the bullet using Newton's second law of motion:
Force (F) = mass (m) × acceleration (a)
50,000 N = 0.014 kg × a
a = 50,000 N / 0.014 kg
a ≈ 3.57 × 10^6 m/s²
Now, we can use the equation of motion to find the distance traveled by the bullet until it comes to rest:
v² = u² + 2as
Where:
v = final velocity (0 m/s)
u = initial velocity (306 m/s)
a = deceleration (-3.57 × 10^6 m/s²)
s = distance traveled (unknown)
Plugging in the values, we get:
0 = (306 m/s)² + 2 × (-3.57 × 10^6 m/s²) × s
Simplifying the equation, we have:
0 = 93,636 - 7,14 × 10^6s
7,14 × 10^6s = 93,636
s = 93,636 / (7,14 × 10^6)
s ≈ 0.0131 m
Therefore, the bullet will penetrate the block of wood approximately 0.0131 meters (or 13.1 mm).