Understanding collisions, impulse, and conservation of momentum
Momentum is a fundamental concept in physics that describes the quantity of motion of an object. It is closely related to Newton's laws of motion and provides a powerful framework for analyzing collisions and other interactions between objects. Understanding momentum helps us explain everything from car crashes to rocket propulsion.
Linear momentum is defined as the product of an object's mass and velocity. It is a vector quantity, meaning it has both magnitude and direction.
The larger an object's mass or speed, the greater its momentum. This explains why a heavy, slow-moving truck can have more momentum than a fast-moving tennis ball.
Calculate the momentum of a 1,500 kg car traveling at 20 m/s.
\[\vec{p} = m\vec{v} = 1,500 \text{ kg} \times 20 \text{ m/s} = 30,000 \text{ kg⋅m/s}\]
The law of conservation of momentum states that in a closed system (no external forces), the total momentum before an event equals the total momentum after the event. This is a fundamental principle in physics and is particularly useful for analyzing collisions.
A 5 kg object moving at 3 m/s collides with a stationary 2 kg object. After the collision, the 5 kg object moves at 1 m/s. What is the velocity of the 2 kg object?
Using conservation of momentum:
\[m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}\]
\[(5)(3) + (2)(0) = (5)(1) + (2)v_{2f}\]
\[15 = 5 + 2v_{2f}\]
\[v_{2f} = 5 \text{ m/s}\]
Impulse is the product of a force and the time interval over which it acts. It is equal to the change in momentum it produces.
The impulse-momentum theorem is a reformulation of Newton's second law and shows that a force acting over time changes an object's momentum.
A 0.15 kg baseball is pitched at 40 m/s. A batter hits the ball, and it travels at 60 m/s in the opposite direction. If the bat is in contact with the ball for 0.002 seconds, what is the average force exerted by the bat?
Change in momentum: \(\Delta p = m(v_f - v_i) = 0.15 \times ((-60) - 40) = -15\) kg⋅m/s
Average force: \(F = \frac{\Delta p}{\Delta t} = \frac{-15}{0.002} = -7,500\) N
The negative sign indicates the force is opposite to the original motion.
Collisions are interactions where objects come into contact and exert forces on each other for a short time. They can be classified into three types:
In elastic collisions, both momentum and kinetic energy are conserved. These collisions are idealized but closely approximated by collisions between hard objects like billiard balls.
In inelastic collisions, momentum is conserved but kinetic energy is not. Some kinetic energy is converted to other forms (like heat, sound, or deformation).
In perfectly inelastic collisions, the objects stick together after collision and move with a common velocity.
A 2000 kg car moving at 15 m/s collides with a stationary 1500 kg car. If they stick together after the collision, what is their common velocity?
\[v_f = \frac{m_1v_{1i} + m_2v_{2i}}{m_1 + m_2} = \frac{(2000)(15) + (1500)(0)}{2000 + 1500}\]
\[v_f = \frac{30,000}{3500} = 8.57 \text{ m/s}\]
The center of mass is the point that represents the average position of all the mass in a system. For a system of particles, the center of mass moves as if all the mass were concentrated at that point and all external forces were applied there.
The momentum of a system can be written in terms of the center of mass motion:
Understanding momentum is essential for analyzing many real-world phenomena and designing various technologies: