An apparatus for measuring the muzzle velocity of a rifle is shown in the figure
ID: 1637898 • Letter: A
Question
An apparatus for measuring the muzzle velocity of a rifle is shown in the figure. A bullet is fired into a 4.500-kg wooden block resting on a smooth surface, and attached to a spring of spring constant k =143 N/m. The bullet, mass 6.9 g, remains embedded in the wooden block. The maximum distance that the block compresses the spring is 9.50 cm. What is the speed V of the bullet? A "seconds" pendulum has a period of exactly 2.000 s. That is, each one-way swing takes 1.000 s. What is the length of a "seconds" pendulum in Austin, Texas, where g=9.793 m/s^2 If the pendulum is moved to the moon, where g=1.620 m/s^2, what must be the length of the pendulum? The position of a SHO as a function of time is given by x = 3.5 cos (7 pi/4 t + pi/4) where t is in seconds and x in meters. Find the (a) period, and (b) the position and velocity at t =0.
Explanation / Answer
Since the collision between the bullet and the block is inelastic, the sum of kinetic and potential energy is not conserved. Some of it ends up as heat in the block and the bullet. Momentum will be conserved though. At the instant the bullet hits the block it momentum is transferred to the bullet block combination.
so,
let's assume the initial velocity of the bullet is u.
applying the principle of conservation of linear momentum,
mu = (M+m)V
or,
(0.0069)(u) =(4.5+0.0069)(v)
or the final velocity of the system = 0.00153 u ----------(1)
The kinetic energy of the block-bullet combination is (1/2)*(4.5069)*(v)2. = (2.253)*(v)2------(2)
The spring will compress until this kinetic energy is transferred into potential energy of the spring. The potential energy of a spring is given by pe = 1/2*k*x2 where x is the amount of compression.
so, P.E. of spring is = (1/2)(143)(0.095 )2 = 0.6452 J ------------(3)
equating equation 2 and 3, we get.
0.6452 = (2.253)*(v)2
substituting the value of v from equation 1, we get
0.6452 = (2.253)*(0.00153 u)2
u = 349.76 m/s
so the velocity of the bullet is 349.76 m/s or appx 350 m/s
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