2017 NYJC P1 Q10

A car is travelling at a speed of \(30\)m/s on a road heading towards a perpendicular train track, which is elevated \(30\) m above the ground. The front of the car is \(40\) m away form the track when the front o the train first crossed the road.

If the train is travelling at \(20\)m/s, show that the distance between the front of the train and the car is \(\sqrt{1300{{t}^{2}}-2400t+2500}\) m.[2]

1. How fast is the front of both the train and the car separating \(1\) second later?[2]
2. Find the distance when the front of the train and the front of the car are closest.[4]
3. Find the rate of change of the angle of elevation of the front of the train from the car \(1\) second later.[4]