Please show all work and round to 4 decimal places if possible. Thanks! You are
ID: 1502074 • Letter: P
Question
Please show all work and round to 4 decimal places if possible. Thanks!
You are driving North through an intersection in a 55 mi/hr speed zone, when the local Chief of Police, who is driving his new Cadillac and approaching the intersection from the West, hits you broadside. The two cars stick together and skid a distance 45.2 m with locked wheels at an angle of 67.2 degree to the East of North. The mass of your car is 1509.0 kg while the Cadillac has a mass 1968.0 kg. The coefficient of sliding friction is 0.60. The Chief of Police is angry that you have damaged his new Cadillac and gives you a ticket for speeding. The local judge is going to believe his Chief of Police rather than some out-of-town student. You realize that the knowledge you learned in your physics course is your only hope for acquittal. Compute the speed of the Chief of Police immediately prior to the collision. Compute your speed immediately prior to the collision as evidence for or against the speeding ticket.Explanation / Answer
Use conservation of momentum. If we assume the collision was instant, the impulse of friction is negligible compared to the collision impulses. Assume also that there is no loss of mass.
Call yourself car A, and the police chief's Cadillac car B.
Define states:
1: before collision
2: immediately after collision
3: after coming to rest after collision
Use conservation of momentum to relate state 1 to state 2. Use the work-energy theorem to relate state 2 to state 3.
Define north as +y, and east as +x.
Your initial velocity: v_a1 = <0, Va>
His initial velocity: v_b1 = <Vb, 0>
Final combined velocity: theta is the angle of the skid east of north
v2 = V2*<sin(theta), cos(theta)>
Your initial momentum: p_a1 = <0, ma*Va>
His initial velocity: v_b1 = <mb*Vb, 0>
Total initial momentum:
p1 = <mb*Vb, ma*Va>
Final momentum: when
p2 = (ma+mb)*v2*<sin(theta), cos(theta)>
Kinetic Energy at state 2:
KE2 = 1/2*(ma + mb)*v2^2
Kinetic Energy at state 3:
KE3 = 0, because you slow down to rest
Heat dissipated by friction:
Q = mu*N*d
N is the normal force, equal and opposite to weight in order that neither car sinks through the asphalt.
Therefore:
Q = mu*d*g*(ma + mb)
Work-energy theorem:
KE2 = Q
1/2*(ma + mb)*v2^2 = mu*d*g*(ma + mb)
Solve for v2:
v2 = sqrt(2*mu*d*g)
Return to conservation of momentum, creating our two equations to solve for two unknowns.
<mb*Vb, ma*Va> = (ma+mb)*v2*<sin(theta), cos(theta)>
Expand out:
mb*Vb = (ma+mb)*v2*sin(theta)
ma*Va = (ma+mb)*v2*cos(theta)
Solve for Va and Vb, the pre-collision speeds:
Vb = v2*sin(theta)*(ma+mb)/mb
Va = v2*cos(theta)*(ma+mb)/ma
Recall expression for v2:
Vb = sqrt(2*mu*d*g)*sin(theta)*(ma+mb)/mb
Va = sqrt(2*mu*d*g)*cos(theta)*(ma+mb)/ma
Data:
mu:=0.6; theta:=67.2 deg; d:=45.2 m; g:=9.8 N/kg; ma:=1509 kg; mb:=1968 kg;
Results:
Your speed: Va = 20.58 meters/second
His speed: Vb = 37.55 meters/second
Translated to miles per hour in order to compare with the speed limit:
Va = 46.036 mph
Vb = 83.996 mph
Conclusion: Your speed was 9 miles/hr below the speed limit. You do not deserve a ticket. You deserve an insurance claim reimbursement.
His speed was 29 mph over the limit. He is clearly at fault for this accident.
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