A 27 g bullet traveling at 297 m/s is fired into a 0.468 kg wooden block anchored to a 100 N/m spring. How far is the spring compressed
(a) Use conservation of momentum to get the velocity of bullet&block (together) just before the block starts to move. Call that velocity V'
mV = (M+m)V'
(b) Use (1/2)(M+m)V'^2 = (1/2) k X^2
M = 0.468 kg
m = 0.027 kg
k = 100 N/m
(Energy is conserved as the spring is comoressed).
Solvee for X
Thank you very much
To find the distance the spring is compressed, we need to use the principle of conservation of momentum and the equation for the potential energy stored in a spring. Here's how you can solve it:
1. Start by finding the initial momentum of the bullet before it hits the wooden block. Momentum (p) is calculated by multiplying the mass (m) of an object by its velocity (v). In this case, the mass of the bullet is 27 g, which is equivalent to 0.027 kg, and its velocity is 297 m/s. So, the initial momentum of the bullet is:
p_initial = m * v = 0.027 kg * 297 m/s
2. The bullet hits the wooden block and the two objects combine together. Using the principle of conservation of momentum, the momentum after the collision is equal to the initial momentum. Since the bullet and the block move together after the collision, we can calculate the combined momentum as follows:
p_combined = (mass of bullet + mass of block) * final velocity
3. Now, we need to find the final velocity. Since there is no external force acting on the system after the bullet hits the block, we can use the principle of conservation of mechanical energy. Both the bullet and the block experience a change in kinetic energy, which is converted into potential energy stored in the spring. The kinetic energy of an object is given by the equation:
KE = (1/2) * m * v^2
4. The potential energy stored in the spring is given by:
PE = (1/2) * k * x^2
Remember that the potential energy depends on the compression of the spring (x), and the spring constant (k) is given as 100 N/m.
5. Equating the initial kinetic energy (KE_initial) and the final potential energy (PE_final), we can write:
KE_initial = PE_final
(1/2) * m * v^2 = (1/2) * k * x^2
6. Rearranging the equation to solve for the compression of the spring (x), we get:
x = √((m * v^2) / k)
7. Now, substitute the values into the equation:
x = √((0.027 kg * (297 m/s)^2) / 100 N/m)
8. Calculate the value of x to find the compression of the spring.