A-level Physics/Forces, Fields and Energy/Further dynamics

From last year, you should remember kinematics and dynamics, the branch of physics that relates to the motion of objects. We will now expand on this and have a look at what happens when two objects collide, the concept of momentum, and we will take a closer look at Newton's three laws of motion.

Momentum
If you have seen collisions involving two objects, you may have noticed that the velocity of one object seems to be passed to the other object. You may also have noticed that heavier objects seem to pass more velocity on to smaller objects, whereas smaller objects seem to pass less velocity to more massive ones.

What is in fact happening is that momentum is being conserved. Momentum is the product of an objects mass and velocity, or $$p=mv$$. This means that, after a collision, an object that is heavier will have a lower velocity than a lighter object in its place, and vice versa. Momentum is conserved for all collisions. The principle of the conservation of momentum states that:

Within a closed system, the total momentum in any specified direction remains constant.

Momentum is a vector quantity and has the units $$kg\,m\,s^{-1}$$ or $$Ns$$ (Newton-seconds) in the SI system.

Collisions
Since momentum is conserved, the momentum before a collision is equal to the momentum after a collision. You can use this fact to solve problems involving collisions.

$$ \begin{alignat}{2} Before & =After\\ m_1u_1+ m_2u_2 & =m_1v_1 + m_2v_2\\ \end{alignat} $$

For instance, a ball is moving at 3 m/s with mass 3 kg. It hits another ball with mass 1 kg moving at 2 m/s; the two balls collide and the second ball rebounds at 4 m/s. Find the velocity at which ball 1 is moving:

$$ \begin{alignat}{2} Before & = After \\ m_1u_1 + m_2u_2 & = m_1v_1 + m_2v_2 \\ 3\times 3 + 2\times 1 & = 3v  + 1\times 4 \\ 11 & = 3v  + 4 \\ 11-4 & = 3v \\ 7 & = 3v \\ v & = \frac{7}{3} \\ \end{alignat}$$

So the velocity at which ball 1 is moving after the collision is 2.3 m/s (7/3)m/s 1

Newton's first law of motion
A body stays at rest or continues to move at constant velocity unless a resultant force acts on it

Newton's second law of motion
Originally, you learnt this to be:

For an object with constant mass, its acceleration is proportional to the force producing the acceleration, and is in the direction of the force.

However, since you now know that a force changes the rate of change of momentum of an object, we can use a more accurate interpretation of Newton's second law:

The rate of change of momentum of a body is proportional to the resultant force acting on the body and takes place in the direction of the resultant force.

Newton's third law of motion
If body A exerts a force on body B, then body B will exert a force of the same type that is equal in magnitude and opposite in direction on body A..