Recoil and explosion problems are solved with conservation of momentum. The key idea is that the forces separating the
pieces are internal forces. Internal forces can give different pieces large velocities, but they cannot change the total
momentum of the whole system.
Momentum conservation
If the external impulse during the recoil or explosion is negligible, then:
Vector momentum conservation.
\[
\sum \vec p_{\text{before}}=\sum \vec p_{\text{after}}
\]
If the original object of total mass \(M\) has initial velocity \(\vec V\), and it separates into fragments with masses
\(m_i\) and velocities \(\vec v_i\), then:
\[
M\vec V=\sum m_i\vec v_i
\]
Here \(M=\sum m_i\), assuming no mass is lost from the modeled system.
1D recoil
In one dimension, choose a positive direction. Momentum conservation becomes:
\[
MV_x=\sum m_i v_{i,x}
\]
For a gun firing a bullet from rest:
\[
0=m_bv_b+m_Rv_R
\]
Solving for rifle recoil speed:
\[
v_R=-\frac{m_bv_b}{m_R}
\]
The negative sign means the rifle recoils opposite the bullet direction.
2D recoil and explosions
In two dimensions, conserve momentum separately in the \(x\)- and \(y\)-directions:
\[
P_{i,x}=P_{f,x},
\qquad
P_{i,y}=P_{f,y}
\]
That means:
\[
MV_x=\sum m_i v_{i,x},
\qquad
MV_y=\sum m_i v_{i,y}
\]
The two component equations are independent. Solve \(x\) and \(y\) separately, then combine them as a vector.
Solving an unknown fragment velocity
If all fragment velocities are known except one, isolate the unknown fragment momentum:
\[
m_u\vec v_u=M\vec V-\sum_{\text{known}}m_i\vec v_i
\]
Therefore:
\[
\vec v_u=
\frac{M\vec V-\sum_{\text{known}}m_i\vec v_i}{m_u}
\]
In 2D:
\[
v_{u,x}=
\frac{MV_x-\sum_{\text{known}}m_i v_{i,x}}{m_u}
\]
\[
v_{u,y}=
\frac{MV_y-\sum_{\text{known}}m_i v_{i,y}}{m_u}
\]
Energy in explosions
Momentum is conserved in explosions, but kinetic energy is usually not conserved. An explosion transforms internal,
chemical, elastic, or nuclear energy into kinetic energy.
\[
\Delta K=K_f-K_i
\]
where:
\[
K_i=\frac12MV^2
\]
\[
K_f=\sum \frac12m_iv_i^2
\]
If \(\Delta K>0\), kinetic energy has been released from internal energy. If \(\Delta K<0\), kinetic energy has
decreased, so the event is not an ideal explosion; energy was absorbed or dissipated.
Worked example: rifle recoil
A \(0.01\ \mathrm{kg}\) bullet leaves a \(4.0\ \mathrm{kg}\) rifle at \(800\ \mathrm{m\,s^{-1}}\).
The rifle and bullet are initially at rest. Take the bullet direction as positive.
Initial momentum.
\[
P_i=0
\]
Final momentum.
\[
P_f=m_bv_b+m_Rv_R
\]
Use conservation of momentum.
\[
\begin{aligned}
0&=m_bv_b+m_Rv_R\\
v_R&=-\frac{m_bv_b}{m_R}\\
&=-\frac{(0.01)(800)}{4.0}\\
&=-2.0\ \mathrm{m\,s^{-1}}
\end{aligned}
\]
The rifle recoils at \(2.0\ \mathrm{m\,s^{-1}}\) backward.
Worked example: 2D fragment explosion
Suppose a stationary object breaks into three fragments. The first two fragment momenta are known. The third fragment
must carry the opposite total momentum so that the final total remains zero.
\[
\vec p_3=-(\vec p_1+\vec p_2)
\]
Then:
\[
\vec v_3=\frac{\vec p_3}{m_3}
\]
This is why fragments in an explosion often fly in different directions: their vector momenta must add to the original
system momentum.
Formula summary
| Concept |
Formula |
Meaning |
| Total mass |
\(M=\sum m_i\) |
Mass of the original system |
| Momentum conservation |
\(M\vec V=\sum m_i\vec v_i\) |
Total momentum before equals total momentum after |
| 1D recoil |
\(MV_x=\sum m_iv_{i,x}\) |
Momentum balance along one axis |
| 2D x-component |
\(MV_x=\sum m_iv_{i,x}\) |
Horizontal momentum balance |
| 2D y-component |
\(MV_y=\sum m_iv_{i,y}\) |
Vertical momentum balance |
| Unknown fragment velocity |
\(\vec v_u=\frac{M\vec V-\sum_{\text{known}}m_i\vec v_i}{m_u}\) |
Velocity required to conserve momentum |
| Initial kinetic energy |
\(K_i=\frac12MV^2\) |
Kinetic energy before separation |
| Final kinetic energy |
\(K_f=\sum \frac12m_iv_i^2\) |
Kinetic energy after separation |
| Kinetic energy change |
\(\Delta K=K_f-K_i\) |
Energy released or absorbed during recoil/explosion |
Recoil is not a new force law. It is simply momentum conservation applied to a system pushed apart by internal forces.
