Question

A motorist traveling at a constant speed of 90 km/h in a 50 km/h speed zone passes a parked police car. Three seconds after the car passes, the police car starts off in pursuit. The policeman accelerates at 2 m/s2 up to a speed of 30 m/s. and then continues at this speed until he overtakes the speeding motorist. How long from the time he started does it take the police car to overtake the motorist? The motorist continues at a constant speed during this process.

Answer 1

Multiple Choice Answers:
A 55 s
B 75 s
C 45 s
D 37.5 s
E 60

Answer 2

First convert either km/h to m/s or m/s to km/h. I personally prefer to use a method which allows me to cancel off units, like how fractions cancel out in multiplication. For example, I can cancel out and 2s in this example:
\[\frac{1}{2}*\frac{2}{2}=\frac{1}{2}\]

Also, recall that multiplying 1 by anything is still that anything.
And 1km=1000m so
\[\frac{1000m}{1km}=1\]I chose to divide this in such a way that I cancel out the unit that was given. Also,
\[\frac{1h}{3600s}=1\]

So I convert the policeman's final speed to km/h to compare with the car:
\[\frac{90km}{s}=\frac{90km}{s}*1=\frac{90km}{s}*\frac{1h}{3600s}\frac{1000m}{1km}=\frac{25m}{s}\]
Wow, look who is speeding now! (police went 30m/s)

For overtaking, the policeman must cover the distance the dude speeding car covers. This means that:
\[Total\:distance\:travelled\:by\:car=total\:distance\:travelled\:by\:police\]
\[(Car\:Speed)*(Total\:time)=(distance\:covered\:during\:acceleration)+(distance\:during\:constant\:speed)\]

Time the police car under went during accelerating is:
\[v=u+at \rightarrow 30+0+2t \rightarrow t= 15s\]

Distance during this is simply given by using:
\[x=ut+\frac{1}{2}at^{2}\]
Which becomes
\[x=\frac{1}{2}*2*15^{2}=225m\]

I assume I know the time the police went at a constant speed of 30m/s, and I let this time by T.

Thus, the whole eqaution to solve is:
\[25*(3+15+T)=225+30T\]
\[T=45s]
But we must remember, when he started he had to speed up, or accelerate, and this time is 15s. So 45+15=60.

Answer 3

Conversion should be written as 90km/h, not 90km/s

Answer 4

thank you again @Festinger