Year: 1987
Paper: 2
Question Number: 15
Course: LFM Stats And Pure
Section: Geometric Probability
No solution available for this problem.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A train of length $l_{1}$ and a lorry of length $l_{2}$ are heading
for a level crossing at speeds $u_{1}$ and $u_{2}$ respectively.
Initially the front of the train and the front of the lorry are at
distances $d_{1}$ and $d_{2}$ from the crossing. Find conditions
on $u_{1}$ and $u_{2}$ under which a collision will occur. On a
diagram with $u_{1}$ and $u_{2}$ measured along the $x$ and $y$
axes respectively, shade in the region which represents collision.
Hence show that if $u_{1}$ and $u_{2}$ are two independent random
variables, both uniformly distributed on $(0,V)$, then the probability
of a collision in the case when initially the back of the train is
nearer to the crossing than the front of the lorry is
\[
\frac{l_{1}l_{2}+l_{2}d_{1}+l_{1}d_{2}}{2d_{2}\left(l_{2}+d_{2}\right)}.
\]
Find the probability of a collision in each of the other two possible
cases.