Final answer:
The recoil speed of the rifle, when a bullet is fired, can be calculated using the principle of conservation of momentum. In this case, we need to calculate the recoil velocity of the rifle when a 4.9-g bullet is fired with a speed of 250 m/s. Using the given values and the equations of momentum, we can find that the recoil speed of the rifle is approximately 0.220 m/s.
Step-by-step explanation:
This question involves the concept of conservation of momentum. According to the principle of conservation of momentum, the total momentum before firing the bullet should be equal to the total momentum after firing the bullet. When the bullet is fired, it acquires a forward momentum. To conserve momentum, the rifle must have an equal and opposite momentum in the backward direction. We can use the equations of momentum to solve this problem.
Given:
- Mass of the bullet (m) = 4.9 g = 0.0049 kg
- Initial velocity of the bullet (u) = 0 m/s
- Final velocity of the bullet (v) = 250 m/s
- Mass of the rifle (M) = 35 N
- Recoil velocity of the rifle (V)
Using the principle of conservation of momentum:
(m * u) + (M * U) = (m * v) + (M * V)
Since the bullet is fired from rest and there is no initial velocity (u = 0):
0 + (35 N * U) = (0.0049 kg * 250 m/s) + (35 N * V)
0 + (35 N * U) = 1.225 kg * m/s + (35 N * V)
35 N * U - 35 N * V = 1.225 kg * m/s ----(1)
We know that weight (W) = Mass (m) * Acceleration due to gravity (g)
or W = m * g
g = W / m = 35 N / 35 kg = 1 N
Therefore, acceleration due to gravity (g) = 1 N
So, the equation (1) can be rewritten as:
35 N * U - 35 N * V = 1.225 kg * m/s^2 ----(2)
We need one more equation to solve this problem. The equation for impulse can be used:
Impulse (J) = Force (F) * Time (t)
Impulse experienced by the bullet = impulse experienced by the rifle
J_b = J_r
or F_b * t_b = F_r * t_r
The force (F) can be calculated as:
F = m * a
where m is the mass and a is the acceleration
Acceleration (a) = Final velocity (v) / Time (t)
So, the equation can be rewritten as:
m_b * v_b / t_b = m_r * v_r / t_r
m_b * v_b * t_r = m_r * v_r * t_b
0.0049 kg * 250 m/s * t_r = 35 kg * V * t_b
t_b = (0.0049 * 250) / (35 * V) = 0.035 / V
Substituting the value of t_b in the equation (2), we get:
35 N * U - 35 N * V = 1.225 kg * m/s^2
35 N * U - (1.225 kg * m/s^2) / (0.035 / V) = 35 N * V
Simplifying the equation:
35 N * U - 35 N * V = (1.225 kg * m/s^2 * V) / (0.035)
35 N * U * 0.035 - 35 N * V * 0.035 = 1.225 kg * m/s^2 * V
1.225 kg * m/s^2 * V = 35 N * U * 0.035 - 35 N * V * 0.035
(1.225 kg * m/s^2 + 35 N * 0.035) * V = 35 N * U * 0.035
V = (35 N * U * 0.035) / (1.225 kg * m/s^2 + 35 N * 0.035)
V = 35 N * U * 0.035 / (1.225 kg * m/s^2 + 35 N * 0.035)
Calculating the value of V, we get:
V ≈ 0.220 m/s
Therefore, the recoil speed of the rifle is approximately 0.220 m/s.