If a 696 N man holds the rifle firmly against
his shoulder, find the recoil speed of man and
rifle.
Answer in units of m/s.
Assuming this is a continuation of your other rifle problem:
Mass of the man + rifle: 696N + 17N = 713N
m = 713N/9.80 m/s^2 = 72.8 kg
(3.6 * 10^-3 kg)/72.8 kg * (231 m/s)=
0.01142 m/s or 1.1 * 10^-2 m/s
To find the recoil speed of the man and the rifle, we need to apply the principle of conservation of momentum. In this case, we can assume that the man and the rifle form an isolated system, meaning that no external forces are acting on them.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant before and after an event. In this case, the event is the man firing the rifle.
The momentum of an object is defined as the product of its mass and velocity: momentum = mass × velocity.
Before firing the rifle, the man and the rifle are at rest. Therefore, their initial momentum is zero.
Let's assign variables for the unknowns:
- Mass of the man = m1
- Mass of the rifle = m2
- Velocity of the man after firing = v1
- Velocity of the rifle after firing = v2
The total momentum before the firing event is given by:
Initial momentum = m1 × 0 + m2 × 0 = 0
After the firing event, the man and the rifle will have a combined momentum in the opposite direction of the bullet's ejection. Since the question does not provide the mass of the bullet or its velocity, we can ignore it in this problem.
The total momentum after the firing event is given by:
Final momentum = m1 × v1 + m2 × v2
Since the total momentum of the system is conserved, we can equate the initial and final momentum expressions:
0 = m1 × v1 + m2 × v2
Now, let's plug in the given information:
- m1 = mass of the man (unknown)
- m2 = mass of the rifle (unknown)
From the question, we know that the force exerted by the rifle (and felt by the man) is 696 N. Force is defined as the rate of change of momentum with time, or F = Δp/Δt.
We can use this force to relate the mass and velocity of the man and the rifle. The force exerted on the system is equal to the rate of change of momentum of the system.
F = Δp/Δt
In this case, we know the force (696 N), but we don't have the change in time (Δt). However, we can make some assumptions to find the unknown variables.
Assumption 1: The man and the rifle are firmly held together, meaning they move as one entity after the firing event. So, v1 = v2 = V (say).
Assumption 2: The time taken for the man and the rifle to reach their final recoil speed is the same. Therefore, v1 and v2 are over the same time interval, which we can cancel out.
Using assumption 1, we can rewrite the equation F = Δp/Δt as F = (m1 + m2) × V, because both the man and the rifle have the same final velocity.
Now we have:
F = (m1 + m2) × V
We know that F = 696 N, so we can substitute this value into the equation:
696 N = (m1 + m2) × V
Now we have one equation (Equation 1) relating the unknowns m1, m2, and V.
Next, we need to find another equation to solve for the unknowns.
The question asks for the recoil speed of the man and the rifle. We can consider the recoil speed of the rifle as the speed of the bullet being ejected, and the recoil speed of the man as the speed at which he moves backward due to the recoil.
This means that the momentum of the bullet and the momentum of the man + rifle system have equal magnitudes but opposite directions.
Since mass × velocity = momentum, we can write the momentum of the bullet as:
p_bullet = -m_bullet × v_bullet
And the momentum of the man + rifle system as:
p_man+rifle = (m1 + m2) × V
From the information given in the question, we can see that the magnitude of the forces exerted on the bullet and the man + rifle system are equal. Mathematically, this can be written as:
|F_bullet| = |F_man+rifle|
Since force is the rate of change of momentum, we can write:
|F_bullet| = |Δp_bullet/Δt| = |Δp_man+rifle/Δt|
Using the assumptions stated earlier, we cancel out the time interval:
|F_bullet| = |p_bullet|/|t| = |p_man+rifle|/|t|
Since |F_bullet| = 696 N:
696 N = |-m_bullet × v_bullet|/|t| = |(m1 + m2) × V|/|t|
We know that v_bullet is the recoil speed of the rifle and can be denoted as V_bullet (say). Therefore, using the equation m_bullet × V_bullet = -m_bullet × v_bullet, we can write:
|m_bullet × V_bullet| = m_bullet × (-v_bullet) = m_bullet × V_bullet
Now we have:
696 N = |(m1 + m2) × V|/|t|
However, we still have two unknowns in this equation, m1 and m2.
To solve for these unknowns, we need to make another assumption:
Assumption 3: The mass ratio of the man to the rifle is given by the equation:
m1/m2 = M (a constant)
Let's substitute the value of M into this equation; for example, let's assume M = 2 (this is just an example—any reasonable value can be used):
m1/m2 = 2 (Equation 2)
Now we have Equation 1 and Equation 2. These two equations can be solved simultaneously to find the values of m1, m2, and V.
Let me calculate it for you.