Momentum, Impulse, Collision

• Momentum is defined as the mass times the velocity.
• The symbol for momentum is p.
• Momentum changes whenever the mass or velocity of an object changes.
• The units of momentum are kg  m/s.
• Momentum is a vector quantity, with the momentum vector pointing in the same direction as the velocity vector.
• The change in momentum of an object is determined by the vector nature of momentum.
• A 1-kg rubber ball with a speed of 4 m/s just before hitting the floor will bounce upward with the same speed, meaning its change in momentum is zero.
• The total momentum of a system of objects is the vector sum of the momentums of all the individual objects.
• Due to the vector nature of momentum, it is possible for a system of several moving objects to have a total momentum that is positive, negative, or zero.
• The product of a force and the time over which it acts is defined as the impulse.
• A small force acting over a long time has the same effect as a large force acting over a short time.
• The units of impulse are the same as the units of momentum, namely, kg  m/s.
• Impulse is a vector that points in the same direction as the force.
• When a force acts on an object, it changes the object's momentum, indicating a connection between impulse and momentum change.
• The connection between impulse and momentum change is revealed through the general form of Newton's second law: F = ma.
• Rearranging this equation, we get F = (m/v)a, where m/v is the change in momentum.
• The relationship between the impulse and momentum change is as follows: F = (m/v)a.
• The figure shows the force exerted on a baseball when struck by a bat, which acts for as little as a thousandth of a second, rising to a peak and then falling to zero.
• A complex force, such as the one acting on a baseball, may be replaced with an average force and the time over which the force acts, facilitating problem solving.
• The figure below shows an example of an essentially elastic collision on the left and an inelastic collision on the right.
• A collision in which the kinetic energy is not conserved is called an inelastic collision.
• The final velocity of cart 1 can be positive, negative, or zero, depending on whether m 1 is greater than, less than, or equal to m 2 .
• In an inelastic collision, the final kinetic energy is less than the initial kinetic energy.
• One-half of the initial kinetic energy is converted into other forms of energy such as sound and heat.
• The velocity of the baseball-catcher's mitt collision can be determined using momentum conservation and kinetic energy conservation.
• The velocity of the person and the ball after the collision in Example 1 can be determined using momentum conservation and kinetic energy conservation.
• In the previous example, the mass doubles and the speed is halved, thus, the final kinetic energy is one-half of the initial kinetic energy.
• Elastic collisions are analyzed using both momentum and kinetic energy conservation.
• The final velocity of cart 2, however, is always positive.
• If the masses of the carts are m 1 and m 2 , respectively, then momentum conservation may be expressed as follows:
• The velocity of the truck immediately after the collision in Example 3 can be determined using momentum conservation.
• An inelastic collision where the colliding objects stick together is referred to as a completely inelastic collision.
• The figure below shows the elastic collision between two air-track carts.
• In an elastic collision, the final kinetic energy of the system is equal to its initial kinetic energy.
• Momentum conservation may be applied to find the speed of the two colliding railroad cars in the previous figure after they stick together.
• A collision in which the kinetic energy is conserved is referred to as an elastic collision.
• Objects that bounce off each other with little deformation—like billiard balls—provide a good approximation to an elastic collision.
• Most everyday collisions are far from elastic.
• A person standing under an umbrella experiences rain, which later turns to hail, and the force required to hold the umbrella upright in the hail may be greater, less than, or equal to the force required to hold it in the rain.
• Momentum is conserved when objects collide, but this does not necessarily mean that kinetic energy is conserved as well.