Acceleration, speed, distance problem...

[Raffi]

A motorcycle enters the freeway 8 seconds before a car following it. The motorcycle upon entering the freeway accelerates to 68 mph in 3 seconds. How fast will the car have to travel in order to catch up with the motorcycle in 1 mile. Keep in mind, the car does 0-60 in 9 seconds. If possible, please show me the formula for solving, as I'm sure the Judge may want to see it.
Let me know if you need anymore numbers.
Thank you

[Raffi]

I've tried to solve this problem numerous times but am stumped by the fact that 2 vehicles are invloved and can't seem to figure out the formula for the car's speed.

[Xamfy19]

Upon entering freeway, do you assume car and motorcycle is accelerating from 0? or they have initial speed?

[Raffi]

Both are starting from zero.

[Xamfy19]

I got 91.36 mph to catch motorcycle.
If this is reasonable, then I will show my work later.

[Raffi]

Thanks, I'd really appreciate it if you could let me know how you came to that figure...

[Xamfy19]

Let's start with motorcycle's acceleration

a1 = (68*5280/3600)/3 = 33.244 ft/s^2
Now, calculate how far within the first 3 seconds the motorcycle traveled.
d1 = 0.5*33.244*9 = 149.6 ft
Calculate the remaining distance before 1 mile = 5280 - 149.6 = 5130.4 ft.
Time to travel 5130.4 ft = 5130.4/99.73 = 51.44 seconds
Total motorcycle traveled time = 51.44 + 3 = 54.44 seconds

Since the car entered hwy 8 seconds later, that left 46.44 seconds for the car to catch up the motorcycle.

From the given information, we have to assume the car accelerated from 0 to a speed at constant acceleration, otherwise there is no solution (or many solutions) to this question.

Let's calculate the acceleration of the car.
a2= 60*5280/3600*9 = 9.77 ft/s^2
which was slower than motorcycle. However, the car keep on accelerating up to a speed. Let's find out the acceleration time (during which the car accelerated)
Let d2 = the distance when the car accelerated,
V = the ultimate speed when the car cruising
t = the time the car accelerated

d2 = 0.5 * 9.77 * t^2
V = 9.77 * t
(5250 - d) / (46.44 - t) = V,

Solve these three equations, you will get t = 13.7 seconds (anothe answer, 79 seconds wasn't right)

V = 9.77 * 13.7 = 133.8 ft/s
= 91.2 mph.

That's how I got this answer.