# Thread: Four Vehicles

1. ## Four Vehicles

A car, a van, a truck and a bike are all travelling tin the same direction on the same road, each at its own onstant speed. At 10 am, the car overtakes the van; at noon, it overtakes the truck; at 2pm it overtakes the bike. At 4 pm the ruck overtakes the bike and, at 6pm, the van overtakes the ruck

a The speed of the car is 120 km/h and the speed of the truck is 80 km/h
i Fin the speeds of the van and the bike
ii Dind the time at which the van overtakes the bike

b Let c and T represent the speeds, in km/h, of the car and the truck, respectively
i Find the speeds of the can and the bike in terms of c and T
ii Show that the time when the van overtakes the bike is the smae, regardless of the speeds of the car and the truck

any help would be great

2. I am not doing the entire problem, I just do part of it.

At 10:00 AM the Van and Car are that the same place. After 2 hours the Car catches up to the Truck. Now, before those two hours the Truck was some distance ahead of the car/van call that $s$. Thus, the distance the car traveled in those two hours is the distance the trucked traveled plus $s$. Thus,
$s+2V_{\mbox{truck}}=2V_{\mbox{car}}$
But we know what the speeds are thus,
$s+2(80)=2(120)$
Thus, $s=80 \mbox{km}$
That means the distance from the car/van at 10:00 to the truck was 80 kilometers.
The problem says that at 6pm (after 8 hours) the van overtakes the truck. That means, the the distance the van traveled from 10:00AM to 6:00PM is the same as the distance the truck traveled plus the distance is was already ahead-which we found to be 80.
Thus,
$8V_{\mbox{van}}=80+8V_{\mbox{truck}}$
Since we know the speed of the truck (divide by eight),
$V_{\mbox{van}}=10+80=90\frac{\mbox{km}}{\mbox{hour }}$
Thus, the speed of the van is 90 kilometers per hour.