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Conservation of Linear
Momentum
Objective In this series of experiments, the conservation of linear momentum and kinetic energy will be tested for different types of collisions.
Equipment List Air track, two air track carts with flags and magnetic inserts, set of masses, two photogates and smart timers, computer, mass scales.
Theoretical Background In the previous lab exercise, conservation of energy was explored. In this lab exercise, another conservation principle, the conservation of momentum, will be explored.
Momentum ๐โ is defined a the product of the mass of an object m and its velocity ๐ฃโ:
๐โ = ๐๐ฃโ (1)
Note that, since velocity is a vector, momentum is also a vector. Conservation of momentum is most useful when considering colliding objects. Momentum being conserved means that the amount of momentum a set of objects has before a collision is the same after the collision. This can be expressed mathematically as
๐โ๐ = ๐โ๐ (2)
where ๐โ๐ is the initial momentum and ๐โ๐ is the final momentum. For momentum to be conserved, no net external force must act on the objects. However, any force that acts between the objects (i.e. internal forces) will not affect momentum conservation, since, by Newtonโs third law, this force must affect both equally in magnitude but in opposite directions. The gravitational attraction between the
masses, for example, would not affect momentum conservation. If, however, the masses were on an inclined plane, the net external force pulling both masses down the plane would cause momentum not to be conserved.
If no net external force acts on the two colliding objects, so that momentum is conserved, then there are two cases for how the objects can collide. In the first case, the objects can collide and bounce off each other while conserving kinetic energy. This type of collision is known as an elastic collision. For this type of collision, combining Equations 1 and 2, the conservation of momentum can be written as,
๐ 1 ๐ฃโ1๐ + ๐ 2 ๐ฃโ2๐ = ๐ 1 ๐ฃโ1๐ + ๐ 2 ๐ฃโ2๐ (3)
where m 1 is the mass of the first object, m 2 is the mass of the second object, ๐ฃโ1๐ is the initial velocity of the first mass, ๐ฃโ2๐ is the initial velocity of the second mass, ๐ฃโ1๐ is the final velocity of the first mass, and ๐ฃโ2๐ is the final velocity of the second
mass. Since kinetic energy is also conserved,
1 2 ๐^1 ๐ฃ1๐
2 ๐^2 ๐ฃ2๐
2 ๐^1 ๐ฃ1๐
2 ๐^2 ๐ฃ2๐
Solving equations (3) and (4) together yields the velocities of m 1 and m 2 after the collision.
๐ฃโ1๐ = ๐๐^1 1 โ๐+๐^2 2 ๐ฃโ1๐ + (^) ๐2๐ 1 +๐^2 2 ๐ฃโ2๐ (5a)
๐ฃโ2๐ = (^) ๐2๐ 1 +๐^1 2 ๐ฃโ1๐ + ๐๐^2 1 โ๐+๐^1 2 ๐ฃโ2๐ (5b)
In the other case, the objects can collide and stick together. This type of collision is known as an inelastic collision. In an inelastic collision, the kinetic energy is not conserved. Instead, some of the initial energy goes into other forms, such as heating the objects. For inelastic collisions, conservation of momentum can be written as,
๐ 1 ๐ฃโ1๐ + ๐ 2 ๐ฃโ2๐ = (๐ 1 + ๐ 2 )๐ฃโ๐ (6)
where ๐ฃโ๐ is the final velocity of the two masses.
For this lab exercise, elastic and inelastic collisions between two carts on an air track will be explored to see if momentum, and kinetic energy, is conserved.
- Repeat Step 1~6 in the previous section for three different cases as listed in Table 1.
Note: Students need to think about how to measure v1f in elastic collisions.
Data Analysis
Inelastic Collisions
- Calculate the percent difference between the theoretical and experimental values of vf using the following equation for theoretical values. ๐ฃ๐ =
- Calculate the initial and final total linear momentum ( Pi and Pf ), as well as the kinetic energies ( KEi and KEf ).
- Plot Pf as a function of Pi for each case. Determine the slope of this line and record it on your data sheet.
Elastic Collisions
- Calculate the percent difference between the theoretical and experimental values of v1f and v2f. Use Equation (5) to determine the theoretical values.
- Calculate the initial and final total linear momentum ( Pi and Pf ), as well as the kinetic energies ( KEi and KEf ).
- Plot Pf as a function of Pi for each case. Determine the slope of this line and record it on your data sheet.
- Plot KEf as a function of KEi for each case. Determine the slope of this line and record it on your data sheet.
Selected Questions
- Suppose the magnetic insert on the carts was replaced with velcro to hold the carts together when they collided. What effect would the velcro have on the conservation of momentum between the two carts? Explain your reasoning.
- Suppose the air track was tilted. What effect would this tilt have on conservation of momentum between the two carts? Explain your reasoning.
- How does friction affect the conservation of momentum in this experiment? Explain your reasoning.
- Suppose magnets were set on two air track carts, mleft=100g and mright=150g, so that they repelled instead of attracted and that the two carts were initially tied together. The two carts were originally moving at a speed of 10.0cm/s to the left, and the final speed of the cart on the left (mleft) is 30.0 cm/s after the string holding the two carts together snaps. What is the speed of the cart on the right (mright) after the string snaps? (Hint: Consider the formula for elastic collisions.)
- Show that, if: (1) the second cart has no initial velocity, (2) the carts collide inelastically, and (3) the carts were of equal mass, then the ratio of the final kinetic energy to the initial kinetic energy is 1/2.
