11-10-25 Energy in Collisions

Collisions

• As we saw in Chapter 3, collisions are situations where we can ignore the interactions between the system and the surroundings. This allowed us to say that:

  • $\Delta \overrightarrow p_{sys} = \overrightarrow F_{net} \Delta t {{\overrightarrow F_{net} = 0}\over \rightarrow} \Delta \overrightarrow p_{sys} = 0$

• Now, we can also say that during a collision:

  • $\Delta E_{sys} = \sum$ Transfers ${\underrightarrow {\sum Transfers = 0}} \Delta E_{sys} = 0$

Elastic Collisions

• During a collision, the size, shape, and other internal properties of a system might change. This change can be temporary or permanent

• When the change is temporary, we say that the collision is an elastic collision.

• For elastic collisions, we have that:

  • $\Delta E_{rest} = 0$ and $\Delta U = 0$ and $\Delta E_{int} = 0$ $\rightarrow \Delta K = 0$

• In other words, momentum and kinetic energy are both conserved in an elastic collision

  • $\overrightarrow p_{sys,f} = \overrightarrow p_{sys,i}$ and $K_f = K_i$

Inelastic Collisions

• When the change of internal properties is permanent, we say that the collision is an inelastic collision.

• For inelastic collisions, we cannot say that kinetic energy remains the same since the other types of energy in the system are changing:

  • $\overrightarrow p_{sys} = \overrightarrow p_{sys,i}$ and $K_f \ne K_i$

• A collision where objects stick together after the collision is sometimes called a maximally inelastic collision.

Q11.x

Which of the following is a property of all elastic collision?

$A)$ The colliding objects interact through springs.

• False — not all elastic collisions literally involve springs; that’s just one model

$B)$ The kinetic energy of one of the objects doesn’t change.

• False — individual object’s kinetic energies can change; only the total is conserved

$C)$ The total kinetic energy is constant at all times — before, during, and after the collision.

• False — during the collision, some kinetic energy may temporarily be sored as potential energy, so the total kinetic energy is not constant at all times

$D)$ The total kinetic energy after the collision is equal to the total kinetic energy before the collision.

• True — in an elastic collision, the total kinetic energy before the collision equals the total kinetic energy after the collision.

$E)$ The spring potential energy after the collision is greater than the spring potential energy before the collision.

• False — the potential energy returns to zero (or to its original value) after the collision.

Q11.x

Two asteroid in deep space are seen to collide. Asteroid 1 has a mass of 7 kg and an initial velocity of <20,0,0> m/s. Asteroid 2 has a mass of 11kg and an initial velocity of <4,0,0> m/s. After the collision, Asteroid 1 has a final velocity of <10,0,0> m/s, but Asteroid 2 is hidden and its velocity could not be measured. Was the collision of the two asteroids elastic or inelastic

$A)$ Elastic

$B)$ Inelastic

$C)$ Unable to tell with the given information

Gather) 

• $m_1 = 7kg$

• $v_{i,1} =$ <20,0,0>

• $m_2 = 11kg$

• $v_{i,2} =$ <4,0,0>

• $v_{f,1} =$ <10,0,0>

• $v_{f,2} =$ $??$

Organize)

• $\overrightarrow p {sys} = \Delta \overrightarrow p_1 + \Delta \overrightarrow p_2$

• $\Delta E = \Delta E_{rest} + \Delta K + \Delta U + \Delta E_{int} = (\Delta K_1 + \Delta K_2) + \Delta E_{int}$

  • $\Delta E_{rest}$ and  $\Delta U$ are negligible

• $\overrightarrow F_{net} = 0$ & $\sum Transfers = 0$

Analyze) 

• $\Delta \overrightarrow p_{sys} = \overrightarrow F_{net} \Delta t \Rightarrow \Delta \overrightarrow p_{sys} = 0 \Rightarrow \Delta \overrightarrow p_1 + \Delta \overrightarrow p_2 = 0$

• $\overrightarrow p_{2,f} - \overrightarrow p_{2,i} = - (\overrightarrow p_{1,f} - \overrightarrow p_{1,i}) \Rightarrow \overrightarrow p_{2,f} = \overrightarrow p_{2,i} - \overrightarrow p_{1,f} + \overrightarrow p_{1,i}$

• $\Delta E_{sys} = 0 \Rightarrow \Delta K_1 + \Delta K_2 + \Delta E_{int} = 0$

• $\Delta E_{int} = - \Delta K_1 - \Delta K_2 = - (\frac 12 m_1 |\overrightarrow v_{1,f}|² - \frac 12 |\overrightarrow v_{1,i}|²) - (\frac {|\overrightarrow p_{2,f}|²}{2m_2} - \frac 12 m_2 |\overrightarrow v_{2,1}|²)$

• $\Delta E_{int} = 547 J$