The expression for the bullet's speed vbullet is
and the speed of a 9.0 g bullet is v=16.8 m/s
<h3>Explanation:
</h3>
A bullet of mass m is fired into a block of mass m that is at rest. the block, with the bullet embedded, slides distance d across a horizontal surface. the coefficient of kinetic friction is μk.
The initial kinetic energy of the bullet is given by
![K_i = \frac{1}{2}mv^2](https://tex.z-dn.net/?f=K_i%20%3D%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
where
m is the mass of the bullet
v is the initial speed of the bullet
The expression for the bullet's speed vbullet is
![F=\mu_k (m+M) g](https://tex.z-dn.net/?f=F%3D%5Cmu_k%20%28m%2BM%29%20g)
where
is the coefficient of kinetic friction
g is the acceleration of gravity
The work done by the force of friction is
![W=-Fd = -\mu_k (m+M)g d](https://tex.z-dn.net/?f=W%3D-Fd%20%3D%20-%5Cmu_k%20%28m%2BM%29g%20d)
where d is the displacement of the block+bullet.
Because the final kinetic energy is zero (the bullet with the block comes at rest), we can write:
And so
![-\mu_k (m+M) g d = -\frac{1}{2}mv^2](https://tex.z-dn.net/?f=-%5Cmu_k%20%28m%2BM%29%20g%20d%20%3D%20-%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
By solving for v, the solution for the bullet speed:
![-\mu_k (m+M) g d = -\frac{1}{2}mv^2\\v=\sqrt{\frac{2\mu_k (m+M)gd}{m}}](https://tex.z-dn.net/?f=-%5Cmu_k%20%28m%2BM%29%20g%20d%20%3D%20-%5Cfrac%7B1%7D%7B2%7Dmv%5E2%5C%5Cv%3D%5Csqrt%7B%5Cfrac%7B2%5Cmu_k%20%28m%2BM%29gd%7D%7Bm%7D%7D)
The speed of a 9.0 g bullet that, when fired into a 12 kg stationary wood block causes the block to slide 5.4 cm across a wood table. Assume that k=0.20.
We have:
the mass of the bullet, m = 9.0 g = 0.009 kg
the mass of the block, M = 12 kg
the distance covered by the block+bullet, d = 5.4 cm = 0.054 m
the coefficient of friction, ![\mu_k = 0.20](https://tex.z-dn.net/?f=%5Cmu_k%20%3D%200.20)
the acceleration of gravity, ![g = 9.8 m/s^2](https://tex.z-dn.net/?f=g%20%3D%209.8%20m%2Fs%5E2)
By substituting, we got
![v=\sqrt{\frac{2 (0.20) (0.009 kg+12 kg)(9.8 m/s^2)(0.054 m)}{0.009 kg}}=16.8 m/s](https://tex.z-dn.net/?f=v%3D%5Csqrt%7B%5Cfrac%7B2%20%280.20%29%20%280.009%20kg%2B12%20kg%29%289.8%20m%2Fs%5E2%29%280.054%20m%29%7D%7B0.009%20kg%7D%7D%3D16.8%20m%2Fs)
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