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