In order to measure the muzzle velocity of a gun, a bullet is fired into a block
ID: 2019504 • Letter: I
Question
In order to measure the muzzle velocity of a gun, a bullet is fired into a block of wood, and the distance the block slides is measured. From this distance one can work backward to determine what the muzzle velocity must have been. Suppose a 15.0-g bullet strikes and becomes embedded in a 1.24-kg block of wood placed on a horizontal surface just in front of the gun. The coefficient of kinetic friction between the block and the surface is 0.28, and the impact drives the block a distance of 11.0 m beforeit comes to rest. (a) Find the velocity of the block immediately after being struck by the bullet. (b) Calculate the muzzle velocity
Explanation / Answer
When the bullet hits the wood, friction exerts a force in the opposite direction. This friction force decelerates the block. We can find the deceleration.
F = ma
F/m = a
F = mgu, where u is 0.28
F = (1.24+.015)kg*9.8m/s^2*0.28 = 3.44 kgm/s^2
a = 3.44 kg m/s^2/1.255kg = 2.744 m/s^2
We know Vf^2 = Vo^2 + 2 a d. Final velocity, Vf = 0. a = -2.744 since it is working to slow down the block.
Vo^2 = 2(2.744m/s^2)(11m) = 60.368m^2/s^2
Vo = 7.77m/s. This is the speed of the block immediately after being hit
(a) 7.77m/s
(b) conservation of momentum will enable us to find muzzle velocity.
VmMr = (Mb+Mr)Vo
Here Vm is muzzle velocity, Mr is the mass of the round, and Mb is the mass of the block.
Vm = (Mb+Mr)Vo/Mr = 1.255*7.769684678/0.015 = 650 m/s
This is a good speed for a rifle. 22s are slower, 5.56 mm NATO rounds are faster
Let's quickly check our work. Start out going about 8m/s, after 1 second going about 5 m/s, after two going about 2 m/s. In the first second the block would travel (8+5)/2 or six and a half. In the second second the block would travel (5+2)/2, or three and a half. and in the last part of a second it would go about another meter. 6 1/2 + 3 1/2 + 1 = 11 meters, so the velocity of the block looks good.
1.245 kg/15g is a little less than 100 and 650 is a little less than 100 times 7.77, so the muzzle velocity also looks good.
Try to always check your work. It's nice when the problem gives real-world values, so try to be familiar with typical values of things as another check of your work.
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