A 4.7 g bullet is fired horizontally into a 0.50 kg block of wood resting on a frictionless table. The block, which is attached to a horizontal spring, retains the bullet and moves forward, compressing the spring. The block-spring system goes into SHM with a frequency of 8.0 Hz and an amplitude of 15 cm. Determine the initial speed of the bullet. (answer should be in m/s)
k=M•ω²=M•(2•π•f)²=0.5(2•π•8)² = 1263.3 N/m
(m+M)•u²/2=k•x²/2
u=x•sqrt(k/(m+M)= 0.15•sqrt(1263.3/0.5047) =7.5 m/s.
m•v=(m+M)•u,
v=(m+M)•u/m =0.5047•7.5/0.0047=805.7 m/s
To determine the initial speed of the bullet, we can use the conservation of momentum principle.
The conservation of momentum principle states that the total momentum before an event is equal to the total momentum after the event, as long as no external forces act on the system.
Before the bullet collides with the block, the bullet is moving horizontally with an initial speed, which we need to find. The block is initially at rest.
After the bullet collides with the block, both the bullet and the block move together as a single system. Since there are no external horizontal forces acting on the system, the total momentum before the collision is equal to the total momentum after the collision.
Let's denote the mass of the bullet as m1, the mass of the block as m2, the initial speed of the bullet as v1, and the final velocity of the block and the bullet together as V.
Initially, only the bullet has momentum, given by:
P1 = m1 * v1
After the collision, the bullet and the block move together, so the total momentum is:
P2 = (m1 + m2) * V
According to the conservation of momentum principle, P1 = P2. Therefore, we have:
m1 * v1 = (m1 + m2) * V
Rearranging this equation, we can solve for v1:
v1 = (m1 + m2) * V / m1
To find V, we need to use the information about the block-spring system in SHM. The frequency of the system (f) can be related to the angular frequency (ω) through the equation:
f = ω / (2 * π)
Where ω = 2 * π * f. Given that the frequency is 8.0 Hz, we can calculate ω.
The period (T) of an object in SHM is the reciprocal of the frequency:
T = 1 / f
Using the value of f, we can find T. The period is also related to the angular frequency through the equation:
T = 2 * π / ω
By equating these two equations and solving for ω, we can find its value.
Once we have ω, we can use the equation for the angular frequency in SHM:
ω = √(k / m_total)
Where k is the spring constant and m_total is the total mass of the block and the bullet.
Solving this equation for k:
k = m_total * ω^2
The amplitude (A) of the SHM is related to the maximum displacement of the block from its equilibrium position:
A = x_max
Given the amplitude as 15 cm, we can convert it to meters.
In SHM, the maximum displacement is related to the spring constant and the total mass through the equation:
x_max = A = √(k / m_total) * (m_total / g)
Where g is the acceleration due to gravity. Solve this equation for m_total:
m_total = (k / (A^2)) * g
With the values of k, A, and g, we can calculate m_total.
Now we have all the necessary values to find V using the equation:
V = √(k / m_total)
Finally, we can substitute the value of V back into the equation for v1 to find the initial speed of the bullet:
v1 = (m1 + m2) * V / m1