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# Teacher Instructions for "A Case Study of Collisions"

As a teacher, you should break the students up into groups of at least three, but there could be more members in a group depending on the class size. When they are separated into groups, have them answer whether or not Pete will go to jail and how fast he was going based on the measurements given.

You will need to provide them with some general physics formulas listed below.

Momentum of inelastic collisions:

pbefore=m1v1+ m2v2pbefore=m1v1+ m2v2
(1)
pafter=m1+ m2vpafter=m1+ m2v
(2)
pbefore= pafterpbefore= pafter
(3)

Friction Force:

Ffriction= μNFfriction= μN
(4)

with N being the normal force

Newton’s Second Law:

F=maF=ma
(5)

Velocity related to displacement:

v2= v02+ 2axv2= v02+ 2ax
(6)

Resultant velocity due to velocity vectors:

vR= vx2+ vy2vR= vx2+ vy2
(7)

Conversions:

## m/s → km/h:

1ms*3600s1hr*1km1000m1ms*3600s1hr*1km1000m
(8)

## km/h → mph:

1kmh=0.62 mph1kmh=0.62 mph
(9)

The students first need to solve the acceleration of the two cars after the collision using equations (4) and (5). Find the force due to friction using the weight of the cars and the coefficient of kinetic friction given in the measurements. This value will be 18110.4 N. Once this is found, set that force equal to mass times acceleration according to Newton’s Second Law, and divide by mass to find the acceleration of the combined cars after the collision. This value will be -6.86 m/s2. Then, using equation (6), they will be able to find the initial velocity (which is also the resultant velocity) of the two cars using acceleration, displacement and knowledge that the final velocity is zero. This value will be 19.48 m/s.

Next the students will have to solve for the resultant velocity in the x-direction using a combination of equations (1), (2) and (3) and some algebra and the knowledge that there is no x-direction velocity for the second car and no y-direction velocity for the first car. The resultant velocity in the x-direction should be 9.276 m/s. Knowing this value and the combined resultant velocity, the students can use equation (7) to find the resultant velocity in the y-direction. This value should be 17.12 m/s. Once this value is known the students can use the same combination of equations (1), (2) and (3) to find the initial velocity in the y-direction. This value should be 30.538 m/s. Using the conversions of equations (8) and (9), the students will get the initial y-velocity of 68.16 miles per hour. This result means that Pete was speeding and he lied under oath in the court room. Those results will send Pete to jail for a long time.

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