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Momentum Your Objectives • You should understand impulse and momentum to relate mass, velocity, and momentum for a moving object or to calculate the total momentum for a system of bodies • You should be able to relate impulse (J) to the change in linear momentum and the average force acting on a body. • You should be able to determine impulse and relate it to change in momentum from Force-Time graphs • You should be able to identify situations in which linear momentum (or its components x&y) is conserved. • You should be able to apply the conservation of momentum to determine final velocity when two objects move along the same line or at right angles, collide, and stick together, and to calculate the change of kinetic energy of such an event. Momentum • A measure of how hard it is to stop a moving object. • Related to both mass and velocity. • Possessed by all moving objects. Calculating Momentum • For one particle p = mv • For a system of multiple particles P = pi = mivi • Momentum is a vector! Which has the most momentum? A brief note about Kinetic Energy • Kinetic energy is the energy of motion Calculating Kinetic Energy (KE) • Kinetic energy is very similar to momentum. • Since an object must be in motion in order to have kinetic energy, VELOCITY is a factor. (It is THE MOST IMPORTANT factor) • A large object in motion will have more energy than a smaller object going the same speed, so MASS will be a factor. KE = ½ x mass x velocity2 KE = ½ mv2 KE is sometimes just referred to as K Linear Momentum: Check for Understanding Linear Momentum: Check for Understanding http://www.bozemanscience.com/apphys-049-momentum Impulse (J) The product of an external force and time, which results in a change in momentum •J = F t •J = P •Units: N s or kg m/s 3. J http://www.bozemanscience.com/ap -phys-050-impulse Impulse (J) F(N) 3000 2000 area under curve 1000 0 0 1 2 3 4 t (ms) 2. Answer: b J Law of Conservation of Momentum pb = pa If the resultant external force on a system is zero, then the vector sum of the momenta of the objects will remain constant. pb = pa • This means that 𝒎𝟏 𝒗𝟏𝒊 + 𝒎𝟐 𝒗𝟐𝒊 = 𝒎𝟏 𝒗𝟏𝒇 + 𝒎𝟐 𝒗𝟐𝒇 Collisions • Collisions are governed by Newton's laws. • Newton’s Third Law tells us that the force exerted by body A on body B in a collision is equal and opposite to the force exerted on body A by body B. Collisions During a collision, external forces are ignored. The time frame of the collision is very short. The forces are impulsive forces (high force, short duration). Collision Types • Elastic (hard, no deformation) – P is conserved, K is conserved • Inelastic (soft; deformation) – P is conserved, K is NOT conserved • Perfectly Inelastic (stick together) – P is conserved, K is NOT conserved Elastic and Inelastic Collisions Elastic and Inelastic Collisions Elastic and Inelastic Collisions 6.4 Elastic and Inelastic Collisions Golf and Momentum Consider the elastic collision between the club head and the golf ball in the sport of golf. Golf and Momentum Forces are on the clubhead and ball are equal and opposite. Golf and Momentum The acceleration of the ball is greater because its mass is smaller. Pool and Momentum Consider the elastic collision between a moving ball and a ball that is at rest in the sport of billiards. Pool and Momentum The balls experience forces which are equal in magnitude and opposite in direction. Pool and Momentum Since the balls have equal masses, they experience equal accelerations. Elastic and Inelastic Collisions: Check for Understanding (Hint: See Special Case b) Elastic and Inelastic Collisions: Check for Understanding Elastic and Inelastic Collisions Example : A rubber ball with a speed of 5.0 m/s collides head on elastically with an identical ball at rest. Find the velocity of each object after the collision. Perfectly Inelastic Collision #1 An 80 kg roller skating grandma collides inelastically with a 40 kg kid as shown. What is their velocity after the collision? Perfectly Inelastic Collisions #2 A train of mass 4m moving 5 km/hr couples with a flatcar of mass m at rest. What is the velocity of the cars after they couple? Perfectly Inelastic Collisions #3 A fish moving at 2 m/s swallows a stationary fish which is 1/3 its mass. What is the velocity of the big fish after dinner? Explosion • When an object separates suddenly, this is the reverse of a perfectly inelastic collision. • Mathematically, it is handled just like an ordinary inelastic collision. • Momentum is conserved, kinetic energy is not. • Examples: – Cannons, Guns, Explosions, Radioactive decay. An object splits apart (EXPLOSION) • Examples: – 2 objects together and moving push off from each other – Explosions • Equations – X direction: • Pb1+2 = p1a + p2a – Y direction • 0 = p1a + p2a – Or vice versa for the equations Recoil Problem #1 A gun recoils when it is fired. The recoil is the result of action-reaction force pairs. As the gases from the gunpowder explosion expand, the gun pushes the bullet forwards and the bullet pushes the gun backwards. The Conservation of Linear Momentum: Check for Understanding The Conservation of Linear Momentum: Check for Understanding Explosion Example • A 200 kg car is rolling at 10 m/s when its fuel tank explodes into 4 pieces. If 40 kg of it goes forward at 25 m/s, 80 kg goes backwards, 30 kg goes to the left at 10 m/s, and a piece goes to the right. Find how much mass is going to the right and how fast it is moving. Also find out how fast the 80 kg piece is moving backwards. Elastic and Inelastic Collisions Example : While standing on skates on a frozen pond, a student of mass 70.0 kg catches a 2.00 kg ball travelling horizontally at 15.0 m/s toward him. a) What is the speed of the student and the ball immediately after he catches it? b) How much kinetic energy is lost in the process? Elastic and Inelastic Collisions: Check for Understanding Elastic and Inelastic Collisions: Check for Understanding Elastic and Inelastic Collisions: Check for Understanding Elastic and Inelastic Collisions: Check for Understanding 6.4 Elastic and Inelastic Collisions: Check for Understanding 6.3 The Conservation of Linear Momentum The Conservation of Linear Momentum Momentum is conserved in an isolated system. The motion in two dimensions may be analyzed in terms of the components of momentum, which is also conserved. If m1 and m2 are equal, they will split apart at a 90 degree angle. Inelastic collisions Elastic and Inelastic Collisions • In inelastic collisions, momentum is conserved but kinetic energy is not. • Collisions like the ones shown here, in which the objects stick together , are called completely inelastic collisions. • The maximum amount of kinetic energy lost is consistent with the conservation of momentum. Elastic and Inelastic Collisions: Check for Understanding Elastic and Inelastic Collisions: Check for Understanding Elastic and Inelastic Collisions Sample Problem Suppose three equally strong, equally massive astronauts decide to play a game as follows: The first astronaut throws the second astronaut towards the third astronaut and the game begins. Describe the motion of the astronauts as the game proceeds. Assume each toss results from the same-sized "push." How long will the game last? Calculate momentum of the balls before and after the collision. 2 m/s 3 m/s 2 kg 0 m/s 8 kg Before 2 kg 50o 8 kg V? After