Go To Concept Tutoring Home www.tutor45.com MOMENTUM AND IMPULSE
Linear Momentum: It is the product of the mass of a moving object and its velocity. Velocity is constant in linear momentum. Momentum is a vector quantity with the direction same as the velocity.
∆t: Time interval over which the force acts (s) ∑F: Average force acting on the object over the time interval ∆t (N)
Momentum and impulse in two dimensions:
Conservation of Momentum: If the net force acting on a system of interacting objects is zero, then the linear momentum of the system before the interaction equals the linear momentum of the system after the interaction. Momentum is conserved both in elastic and inelastic collision. Conservation of Momentum for colliding objects: m_{1}.v_{1} + m_{2}.v_{2}... = m_{1}.v^{'}_{1} + m_{2}.v^{'}_{2}... m_{1: } Mass of the fist object_{ }m_{2}: Mass of the second object_{ }v_{1}: Velocity of the fist object before collision_{ }v_{2}: Velocity of the second object before collision_{ }v^{'}_{1}: Velocity of the first object after collision_{ }v^{'}_{2}: Velocity of the second object after collision Conservation of Momentum in two dimensions for interacting objects have also same rule, and this rule applies for x component of the momentum and y component of the momentum. m_{1}.v_{1x} + m_{2}.v_{2x}...= m_{1}.v^{'}_{1x} + m_{2}.v^{'}_{2x}... (x component of the momentum) m_{1}.v_{1y} + m_{2}.v_{2y}... = m1.v^{'}_{1y} + m_{2}.v^{'}_{2y}... (y component of the momentum)
Conservation of Kinetic Energy: Kinetic energy is conserved in elastic collisions only. Kinetic energy is not conserved in inelastic collisions. For elastic collision: 0.5 m_{1}.v_{1}^{2} + 0.5 m_{2}.v_{2}^{2} = 0.5 m_{1}.v^{'}_{1}^{2} + 0.5 m_{2}.v^{'}_{2}^{2}
Momentum is conserved in all kinds of collisions such as elastic, inelastic and completely inelastic collisions in an isolated system. Kinetic energy is not conserved in inelastic collisions. Total kinetic energy after the collision is different from the total kinetic energy before collision. In completely inelastic collision, the objects stick together and the decrease in total kinetic energy will be maximum. Most of the inelastic collisions, the total kinetic energy after the collision is less than the the total initial kinetic energy before the collisions. However, some inelastic collisions, such as an explosion, the total final kinetic energy of the system is greater than the initial kinetic energy of the system because kinetic energy produced during the explosion. A good example of an inelastic collision is the collisions of tennis balls. If the two bodies stick together
after the collision, the collision is said to be completely
inelastic collision. A bullet embedding itself in a block
of wood is an example of inelastic collision. Another example
for completely inelastic collision would be two cars that crash and
lock bump. A meteorite collides headon with the the Earth is
also completely inelastic collision. Elastic Collision Momentum is conserved in all kinds of collisions such as elastic, inelastic and completely inelastic collisions in an isolated system. Kinetic energy is conserved only in elastic collisions. Total kinetic energy after the collision equals total kinetic energy before the collision. In real life, it is very difficult to produce completely elastic collisions. The only really good examples of elastic collisions are collisions of atomic particles. Collisions between hard steel balls as in the swinging balls apparatus are nearly elastic. The collisions of pool balls, or a superballs are also nearly elastic. The collisions of air molecules with the walls of container are elastic collision. Calculations for Onedimensional Elastic Collision: Equation 1: m_{1}.v_{1} + m_{2}.v_{2} = m_{1}.v^{'}_{1} + m_{2}.v^{'}_{2 } (Momentum is conserved in all kinds of collisions) Equation 2: 0.5 m_{1}.v_{1}^{2} + 0.5 m_{2}.v_{2}^{2} = 0.5 m_{1}.v^{'}_{1}^{2} + 0.5 m_{2}.v^{'}_{2}^{2 } (Kinetic energy is conserved only in elastic collisions) If the collision is elastic, and if the velocity of the second object before collision equals to zero, we can get the following equations by using above equation 1 and equation 2: m_{1}(v_{1} v^{'}_{1}) = m_{2}  v^{'}_{2} m_{1}(v_{1}^{2}  v^{'}_{1}^{2}) = m_{2}.v^{'}_{2}^{2} v_{1} + v^{'}_{1} = v^{'}_{2 } v^{'}_{1} = v_{1} (m_{1 } m_{2}) / (m_{1 }+ m_{2}) v^{'}_{2} = v_{1} (2m_{1}) / (m_{1 }+ m_{2}) If the collision is elastic, but the velocity of the second object before collision does not equal to zero, we should make adjustment to make the velocity of the second object before collision equals to zero. For example: v_{1}
= 5 m/s v_{1}
= 2 m/s After you complete all the calculations, you add 3 m/s to all the velocities.
