You have been asked to design a "ballistic spring system" to measure the speed of bullets. A bullet of mass m is fired into a block of mass M. The block, with the embedded bullet, then slides across a frictionless table and collides with a horizontal spring whose spring constant is k. The opposite end of the spring is anchored to a wall. The spring's maximum compression d is measured.
(a) Find an expression for the bullet's initial speed in terms of m, M, k, and d.
I got this part right [Ans: (d/m)*sqrt(k*(m+M))]
(b) What was the speed of a 5.2 g bullet if the block's mass is 2.0 kg and if the spring, with k = 41 N/m, was compressed by 10 cm?
Correct answer is 174 m/s.
(c) What fraction of the bullet's energy is "lost"?
^How do I solve part (c)? Thanks
To solve part (c), we need to determine the initial kinetic energy of the bullet and compare it to the kinetic energy of the block-bullet system after the collision with the spring.
The initial kinetic energy of the bullet can be calculated using the equation:
KE_bullet = (1/2) * m * v^2,
where m is the mass of the bullet and v is its initial speed. From part (a), we have the expression for the initial speed of the bullet as:
v = (d/m) * √(k * (m + M)).
Substituting this expression into the equation for kinetic energy, we get:
KE_bullet = (1/2) * m * [(d/m) * √(k * (m + M))]^2.
Simplifying the expression:
KE_bullet = (1/2) * m * [d^2/m^2] * (k * (m + M)).
KE_bullet = (1/2) * d^2/m * √(k * (m + M)).
Now, let's consider the kinetic energy of the block-bullet system after the collision with the spring. The maximum compression of the spring, d, occurs when all the initial kinetic energy of the system is converted to potential energy stored in the spring. Therefore, the total energy of the system after compression is solely potential energy stored in the spring.
The potential energy of the compressed spring can be calculated using the equation:
PE_spring = (1/2) * k * d^2.
Now, let's compare the initial kinetic energy of the bullet with the potential energy of the compressed spring.
Fraction of energy lost = (KE_bullet - PE_spring) / KE_bullet.
Substituting the expressions for KE_bullet and PE_spring, we get:
Fraction of energy lost = [(1/2) * d^2/m * √(k * (m + M))] - (1/2) * k * d^2)] / [(1/2) * d^2/m * √(k * (m + M))].
Simplifying the expression further, we get:
Fraction of energy lost = [d^2 - m * k * d^2 / √(k * (m + M)))] / [d^2/m * √(k * (m + M))].
Fraction of energy lost = [d^2 * (1 - m * k / √(k * (m + M)))] / [d^2/m * √(k * (m + M))].
Fraction of energy lost = [1 - m * k / √(k * (m + M))].
Now, to find the fraction of energy lost, we can substitute the given values into this expression and calculate the result.
To solve part (c), we need to calculate the initial kinetic energy of the bullet and compare it to the final kinetic energy after the collision with the spring.
The initial kinetic energy of the bullet can be calculated using the formula KE = 0.5 * m * v^2, where m is the mass of the bullet and v is its initial velocity.
From part (a), we have an expression for the bullet's initial velocity:
v = (d/m) * sqrt(k * (m + M))
Using this expression, we can substitute it into the formula for KE:
KE_initial = 0.5 * m * ((d/m) * sqrt(k * (m + M)))^2
Simplifying the equation,
KE_initial = 0.5 * m * ((d/m)^2 * k * (m + M))
Expanding and simplifying further,
KE_initial = 0.5 * (d^2/m) * k * (m + M)
Now, we need to calculate the final kinetic energy of the bullet after the collision with the spring. We know that the spring compresses by a maximum distance, and as it decompresses, it transfers some of its energy to the bullet.
The maximum potential energy stored in the spring is given by the equation PE = 0.5 * k * d^2, where k is the spring constant and d is the maximum compression.
Since energy is conserved during the collision, this potential energy is transferred entirely to the bullet in the form of kinetic energy.
KE_final = PE = 0.5 * k * d^2
To find the fraction of the bullet's energy lost, we calculate the difference between the initial and final kinetic energy and divide it by the initial kinetic energy:
Energy lost = (KE_initial - KE_final) / KE_initial
Substituting the expressions for KE_initial and KE_final:
Energy lost = (0.5 * (d^2/m) * k * (m + M) - 0.5 * k * d^2) / (0.5 * (d^2/m) * k * (m + M))
Simplifying the equation further,
Energy lost = [(d^2/m) * k * (m + M) - k * d^2] / [(d^2/m) * k * (m + M)]
Now, let's substitute the given values to solve part (c):
m = 5.2 g = 0.0052 kg
M = 2.0 kg
k = 41 N/m
d = 10 cm = 0.1 m
Plugging these values into the equation,
Energy lost = [(0.1^2/0.0052) * 41 * (0.0052 + 2.0) - 41 * 0.1^2] / [(0.1^2/0.0052) * 41 * (0.0052 + 2.0)]
Calculating the expression yields,
Energy lost = 0.993
Therefore, about 99.3% of the bullet's energy is lost during the collision with the spring.
Note: Remember to use consistent units throughout the calculation to avoid errors.
you have the initial bullet KE (from initial speed0.
You have the spring final energy.
energy lost= iniial KE-finalPEspring
fraction=energy.lost/initialKE