Two particles A and B collide. Particle A has mass 2m and velocity 40 m/s, while particle B has mass m and velocity 10 m/s. Calculate their velocities after a perfectly elastic collision and the impul

For a perfectly elastic collision, the total momentum before the collision equals the total momentum after the collision.

$$m_A u_A + m_B u_B = m_A v_A + m_B v_B$$

Given $m_A = 2m$, $u_A = 40 \, \text{m/s}$, $m_B = m$, $u_B = 10 \, \text{m/s}$.

$$(2m)(40) + (m)(10) = (2m)v_A + (m)v_B$$ $$80m + 10m = 2mv_A + mv_B$$ $$90m = m(2v_A + v_B)$$ $$90 = 2v_A + v_B \quad (*)$$

For a perfectly elastic collision, the relative velocity of approach equals the negative of the relative velocity of separation.

$$u_A - u_B = v_B - v_A$$ $$40 - 10 = v_B - v_A$$ $$30 = v_B - v_A \quad (**)$$

From $(**)$, we can express $v_B$ as $v_B = 30 + v_A$. Substitute this into $(*)$:

$$90 = 2v_A + (30 + v_A)$$ $$90 = 3v_A + 30$$ $$60 = 3v_A$$ $$v_A = 20 \, \text{m/s}$$

Now substitute $v_A$ back into $v_B = 30 + v_A$:

$$v_B = 30 + 20$$ $$v_B = 50 \, \text{m/s}$$

The velocities after collision are:

$$\boxed{v_A = 20 \, \text{m/s}, v_B = 50 \, \text{m/s}}$$

Impulse is the change in momentum ($J = \Delta p = mv - mu$).
Impulse on A ($J_A$):

$$J_A = m_A v_A - m_A u_A$$ $$J_A = (2m)(20 \, \text{m/s}) - (2m)(40 \, \text{m/s})$$ $$J_A = 40m - 80m$$ $$J_A = -40m \, \text{N s}$$

Impulse on B ($J_B$):

$$J_B = m_B v_B - m_B u_B$$ $$J_B = (m)(50 \, \text{m/s}) - (m)(10 \, \text{m/s})$$ $$J_B = 50m - 10m$$ $$J_B = 40m \, \text{N s}$$

The impulses are:

$$\boxed{J_A = -40m \, \text{N s}, J_B = 40m \, \text{N s}}$$

Newton's third law states that for every action, there is an equal and opposite reaction. In terms of impulse, the impulse exerted by A on B is equal in magnitude and opposite in direction to the impulse exerted by B on A.
From our calculations:

$$J_A = -40m \, \text{N s}$$ $$J_B = 40m \, \text{N s}$$

We can see that $J_A = -J_B$. This demonstrates that the impulse on body A due to body B is equal in magnitude and opposite in direction to the impulse on body B due to body A, thus obeying Newton's third law of motion.