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Question: A bullet of mass \(10\) g travelling horizontally with a velocity of \(150 m s^{-1}\) strikes a stat...

A bullet of mass 1010 g travelling horizontally with a velocity of 150ms−1150 m s^{-1} strikes a stationary wooden block and comes to rest in 0.03s0.03 s. Calculate the distance of penetration of the bullet into the block. Also calculate the magnitude of the force exerted by the wooden block on the bullet.

Explanation

Solution

Initial and final velocities are given so we will find the acceleration of the bullet by the first equation of motion. With the help of the third equation of motion, we will find the distance travelled by the bullet. With the help of the second law of motion, we will find the value of force.

Complete answer:
Given: initial velocity, u=150ms−1u = 150 m s^{-1}
Mass of bullet =1010 g
Final velocity, v=0v = 0 as bullet finally comes to rest.
Time taken = 0.03s0.03 s.
According to the first equation of motion, we will find the acceleration.
v=u+atv = u +at
  ⟹  0=150+a(0.03)\implies 0 = 150 +a(0.03)
  ⟹  a=−1500.03=−5000ms−2\implies a =\dfrac{-150}{0.03} = -5000 m s^{-2}
(Negative sign indicates that the velocity of the bullet is decreasing.)
According to the third equation of motion, we will find the distance.
v2=u2+2asv^{2} = u^{2} + 2as
  ⟹  0=1502+2(−5000)s\implies 0 = 150^{2} + 2(-5000)s
  ⟹  s=2.25m\implies s = 2.25 m
Hence, the distance of penetration of the bullet into the block is 2.25m2.25 m.
From Newton’s second law of motion:
F=m×aF = m \times a
  ⟹  F=0.01×(−5000)=−50N\implies F = 0.01 \times (-5000) = -50N
The magnitude of the force exerted by the wooden block on the bullet is 50N50 N.

Additional Information:
When a bullet moves with some energy and penetrates into an object, then the bullet loses energy to the object due to conservation of energy and momentum. Finally, the bullet will stop with zero final velocity and before reaching at rest, it travels some distance into the object due to retardation.

Note:
All the values should be in S.I. unit. When the bullet penetrates the wooden block, then the bullet stops after travelling some distance, so we take final velocity is equal to zero. Also, a negative sign of acceleration shows that the velocity is decreasing and force is acting in the opposite direction.