A 14.3 gram bullet embeds itself in a 2.823 kg block on a horizontal frictionless surface which subsequently collides with a spring of stiffness 378 N/m. The maximum compression of the spring is 13.5 centimetres.
a) What was the initial speed of the bullet in m/s?
b) How long did it take for the bullet-block system to come to rest after contacting the spring?
a) Momentum is conserved during the embedding process. Kinetic energy is conserved during the spring compression by a distance X. Do the problem "backwards": first figure out the velocity after embedding but before comporession, using the elastic relationship for this process.
(1/2)*k*Xmax^2 = (1/2)(m+M)v^2
where v is the velocity after the bullet is embedded but before compression.
(378)*(0.135)^2 = 2.837*v^2
v = 1.558 m/s
Now let V be the initial bullet velocity. Apply conservation of momentum to the bullet-embedding process.
0.0143*V = (m+M)*v
0.0143*V = 2.837*1.558
V = 309 m/s
b)(1/4) of the period of the spring-mass system with the bullet added.
time to stop = (P/4)
= (1/4)(2 pi)*sqrt[(m+M/k)]
= (pi/2)*sqrt[2.837/378]
= 0.136 s
For some reason, it says it's the wrong answer.
Hmmm well I've tried it and, for what it's worth, the way that drwls has done it is right.
Yeah you're right, it does work. I must have done something wrong the first time.
Thanks!
I don't get how you got the right answer...?
To solve this problem, we can use the principle of conservation of momentum and the potential energy stored in the compressed spring.
a) To find the initial speed of the bullet, we need to equate the initial momentum of the bullet to the final momentum of the bullet-block system.
The initial momentum of the bullet can be calculated using the formula:
initial momentum = mass of the bullet × initial velocity of the bullet
The final momentum of the bullet-block system is zero since it comes to rest. The final momentum is given by:
final momentum = mass of the block × final velocity of the block
Since the bullet embeds itself in the block, their final velocities are the same. So we have:
mass of the bullet × initial velocity of the bullet = mass of the block × final velocity of the block
Plugging in the values, we get:
(14.3 g) × (initial velocity of the bullet) = (2.823 kg) × (final velocity of the block)
Note: We need to convert the mass of the bullet from grams to kilograms for consistent units.
Converting 14.3 grams to kilograms:
mass of the bullet = 14.3 g = 0.0143 kg
So the equation becomes:
(0.0143 kg) × (initial velocity of the bullet) = (2.823 kg) × (final velocity of the block)
Now we can solve for the initial velocity of the bullet.
b) To find the duration it takes for the bullet-block system to come to rest after contacting the spring, we can use the potential energy stored in the compressed spring.
The potential energy stored in the compressed spring is given by the formula:
potential energy = (1/2) × spring constant × (compression distance)^2
We can equate this potential energy to the initial kinetic energy of the bullet-block system.
The initial kinetic energy can be calculated using the formula:
initial kinetic energy = (1/2) × mass of the bullet-block system × (initial velocity)^2
Since the bullet is embedded in the block, the mass of the bullet-block system is the sum of their masses.
We can then equate the initial kinetic energy to the potential energy stored in the compressed spring, and solve for the initial velocity of the bullet.
With the initial velocity of the bullet determined, we can calculate the final velocity of the block. Since the block comes to rest, the final velocity will be zero.
Now, using the initial and final velocities of the block, we can calculate the time it takes for the block to come to rest by dividing the change in velocity by the acceleration:
time = (final velocity of the block - initial velocity of the block) / acceleration
Since the surface is frictionless, the only force acting on the block is due to the compressed spring, which acts as a restoring force. According to Hooke's Law, the force exerted by the spring is given by:
force = - spring constant × compression distance
By Newton's Second Law (F = ma), the acceleration of the block is given by:
acceleration = force / mass of the block
Substituting the formula for force, we have:
acceleration = (-spring constant × compression distance) / mass of the block
Now, we can substitute the mass of the block, spring constant, and compression distance into the equation to solve for time.