Consider a collision between air-track gliders equipped with spring bumpers so t

Question

Consider a collision between air-track gliders equipped with spring bumpers so that the collision is elastic. As usual, we assume the air track is frictionless. Glider A has a mass of 0.50 kg, and glider B has a mass of 0.30 kg. Before the collision, (Figure 1) , the gliders are moving toward each other, each with a speed of 2.0 m/s. What are the velocities of Aand B after the collision (Figure 2) ?

SET UP A coordinate system was set up with the +x axis to the right, as shown in the figures. Because we’re dealing with a one-dimensional problem, we’ll omit the x subscripts on the velocities, while keeping in mind that they are all x components of velocities.

SOLVE We can use conservation of momentum, along with the relative-velocity relationship, to obtain two simultaneous equations for the two final velocities.

From conservation of momentum,

mAvA,i+mBvB,i(0.50kg)(2.0m/s)+(0.30kg)(?2.0m/s)(0.50)vA,f+(0.30)vB,f===mAvA,f+mBvB,f(0.50kg)vA,f+(0.30kg)vB,f0.40m/s

From the relative-velocity relationship for an elastic collision,

vB,f?vA,f===?(vB,i?vA,i)?(?2.0m/s?2.0m/s)4.0m/s

Solving these equations simultaneously, we obtain,

vA,f=?1.0m/svB,f=3.0m/s (final x components of velocity).

REFLECT Both gliders reverse their directions of motion; A moves to the left at 1.0 m/s and B moves to the right at 3.0 m/s.

To confirm that the collision was elastic, we calculate the initial and final kinetic energies.

Ki==12(0.50kg)(2.0m/s)2+12(0.30kg)(?2.0m/s)21.6J

Kf==12(0.50kg)(?1.0m/s)2+12(0.30kg)(3.0m/s)21.6J

As expected, the total initial kinetic energy before the collision equals the total final kinetic energy after the collision

Part A - Practice Problem:

Suppose we replace the gliders with two different ones. Glider A now has a mass of 0.15 kg and glider B has a mass of 0.50 kg . Both still have an initial speed of 2.0 m/s toward each other before the collision. Find the final velocity of Glider A if the collision is elastic.

Express your answer in meters per second to two significant figures.

Find the final velocity of Glider B if the collision is elastic.

Express your answer in meters per second to two significant figures.

Explanation / Answer

Their relative speed prior to the collision is 4 m/s
so their relative speed after the collision must also be 4 m/s ( elastic collision. )

Motion of the centre of mass.
(2 * 0.15 - 2 * 0.50) / ( 0.15 + 0.50) = - 1.077 m/s

Motion of each relative to the centre of mass.
A 2 + 1.077 = 3.077 to the right
B 2 - 1.077 = 0.923 to the left.

Collide the two. The velocities relative to the centre of mass reverse.
A 3.077 to the left
B 0.923 to the right

Convert to frame of reference.

A 3.077 + 1.077 = 4.154 to the left
B 0.923 - 1.077 = -0.154 to the right

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