1. A bullet of 0.0500 kg is fired into a block of wood. Knowing that the bullet left the gun
with a muzzle velocity of 350. m/s, and the bullet penetrates .15 m into the block of
wood, determine:
a) The average force required to stop the bullet.
b) The impulse exerted by the wood on the bullet.
c) The change in momentum of the bullet
If the block doesn’t move with bullet inside, the magnitude of acceleration (deceleration) of the bullet is
a =v²/2s=350²/2•0.15=408333 m/s².
F=ma= 0.05•408333= 20417 N.
The change in momentum of the bullet
Δp=p2-p1=0-m•v=-0.05•350=-1.4•10^-4 kg•m/s.
The impulse exerted by the wood on the bullet = Δp=1.4•10^-4 kg•m/s.
a) Well, stopping a bullet is no small task, especially for a block of wood. It's like asking a pea to catch a freight train! You'll need some serious force to make it happen. To calculate the average force required, you'll need the time it takes for the bullet to come to a stop. Without that information, I can't give you a precise answer, but just know that it won't be a gentle nudge.
b) Ah, the classic case of "bullet meets wood." When the bullet penetrates the block, it's like an unexpected surprise party! The wood exerts an impulse on the bullet, which can be calculated using the momentum change of the bullet. I hope the wood said "surprise" when it made the impact!
c) Change in momentum is like a teenager's emotions - it can be quite extreme. To find the change in momentum of the bullet, you'll need to know the initial momentum (before entering the block) and the final momentum (after coming to a stop). The difference between the two will give you the change. It's like figuring out the difference between a sprint and a crash into a wall. The numbers might make you wince!
To solve this problem, we need to use the principles of impulse and conservation of momentum. Here's how we can find the answers to each part of the question:
a) The average force required to stop the bullet can be calculated using the impulse-momentum relationship. The impulse is equal to the change in momentum, and the average force is equal to the impulse divided by the time taken to stop the bullet.
b) The impulse exerted by the wood on the bullet can be found using the impulse-momentum relationship. The impulse is equal to the change in momentum of the bullet.
c) The change in momentum of the bullet can be calculated using the equation Δp = mv - mu, where Δp is the change in momentum, m is the mass of the bullet, v is the final velocity, and u is the initial velocity.
Let's calculate these values step-by-step:
Step 1: Convert the given values to SI units
- Mass of the bullet (m): 0.0500 kg
- Initial velocity of the bullet (u): 350. m/s
- Penetration depth (s): 0.15 m
Step 2: Calculate the final velocity of the bullet (v)
Given that the bullet penetrates 0.15 m into the wood, we can assume it comes to rest inside the wood. Therefore, the final velocity of the bullet (v) is 0 m/s.
Step 3: Calculate the average force required to stop the bullet (F)
We can use the equation F = Δp / t, where Δp is the change in momentum and t is the time taken to stop the bullet.
Since the bullet comes to rest inside the wood, we need to determine the time taken to stop the bullet. We can use the equation s = ut + 0.5at^2 to find the time it takes for the bullet to penetrate into the wood.
Given that u = 350. m/s, a = 0 m/s^2 (assumed to be zero since the bullet stops), and s = 0.15 m, we can rearrange the equation to solve for t:
0.15 = (350*t) + (0.5*0*t^2)
0.15 = 350t
t = 0.15 / 350
t ≈ 0.00042857 s
Now we can calculate the average force using F = Δp / t, where Δp = m(v-u):
Δp = 0.0500 kg * (0 m/s - 350. m/s)
Δp = -17.5 kg·m/s (negative sign indicates a change in direction)
F = Δp / t
F = -17.5 kg·m/s / 0.00042857 s
F ≈ -4.084 x 10^4 N (negative sign indicates the force acts in the opposite direction)
Therefore, the average force required to stop the bullet is approximately -4.084 x 10^4 N.
Step 4: Calculate the impulse exerted by the wood on the bullet (J)
The impulse is equal to the change in momentum, which we already calculated as -17.5 kg·m/s.
Therefore, the impulse exerted by the wood on the bullet is -17.5 kg·m/s.
Step 5: Calculate the change in momentum of the bullet (Δp)
The change in momentum of the bullet is equal to the final momentum minus the initial momentum:
Δp = mv - mu
Δp = 0.0500 kg * (0 m/s) - 0.0500 kg * (350. m/s)
Δp = -17.5 kg·m/s
Therefore, the change in momentum of the bullet is -17.5 kg·m/s.
Summary of results:
a) The average force required to stop the bullet is approximately -4.084 x 10^4 N.
b) The impulse exerted by the wood on the bullet is -17.5 kg·m/s.
c) The change in momentum of the bullet is -17.5 kg·m/s.
To solve this problem, we need to use the principles of impulse and momentum.
a) To find the average force required to stop the bullet, we can use the equation for average force:
Average Force = (Change in Momentum) / (Time)
To find the change in momentum, we can use the equation:
Change in Momentum = Final Momentum - Initial Momentum
Since the bullet is brought to rest, the final momentum is 0. The initial momentum can be calculated using the formula:
Initial Momentum = Mass x Initial Velocity
Given the mass of the bullet (0.0500 kg) and the initial velocity (350. m/s), we can plug these values into the equation to find the initial momentum.
Once we have the change in momentum, we can divide it by the time it takes to stop the bullet. Unfortunately, we do not have the time given in the problem statement, so we cannot directly calculate the average force required to stop the bullet.
b) The impulse exerted by the wood on the bullet can be found using the impulse-momentum equation:
Impulse = Change in Momentum
So, in this case, the impulse exerted by the wood on the bullet is equal to the change in momentum of the bullet.
c) The change in momentum of the bullet can be calculated using the same formula as before:
Change in Momentum = Final Momentum - Initial Momentum
Since the bullet is brought to rest, the final momentum is 0. The initial momentum can be calculated using the formula:
Initial Momentum = Mass x Initial Velocity
Given the mass of the bullet (0.0500 kg) and the initial velocity (350. m/s), we can plug these values into the equation to find the initial momentum.
By subtracting the final momentum from the initial momentum, we can determine the change in momentum of the bullet.