If a dune Buggy is traveling for 20 seconds at a speed of 8.5m/s how do I calculate the unknown quantities

1).  the product of the two masses being gravitationally attracted to each other

2).  the distance between their centers of mass

And that's IT.  The gravitational force between them depends on
only those two things, nothing else.

The equation behind this question is:

g = (G*m*M) / (r^2)

g is the gravitational force
G is the Universal Gravitational Constant
M is the mass of one object
and m is the mass of the other constant
r is the distance between the two objects

Mechanical energy in this problem is the sum of the kinetic energy and the potential energy: 5.2+3.5= 8.7 Joules

Use kinematic equation:

vf² = vo² + 2g(xf - xo)

vf = final velocity = 10 m/s
vo = initial velocity = 0 m/s (i.e. at rest)
g = 9.81 m/s²
xf - xo = distance traveled = d

*rearrange equation to solve for d

(vf² - vo²)/(2g) = d

d = ((10 m/s)² - (0 m/s)²)/(2*9.81 m/s²)

d = (100 m²/s²)/(2*9.81 m/s²)

d = 5.097 m

The answer is b, gravity and friction are forces.

We use the kinematics equation:
Vf = Vi + a*t
8 = 0 + 3.6 * t
t=2.222s to reach 8.0 m/s

At that time the train has moved
4.5 m/s * 2.222s = 9.999 m

He travelled (another kinematics equation)
Vf^2 = Vi^2 + (2*a*d)
(8.0)^2 = (0)^2 + (2 * 3.6 * d)
d=8.888 m

The train is 9.999m, the fugitive is 8.888m,
He still needs to travel
9.999-8.888= 1.111m

He needs to cover the rest of the distance in a smaller amount of time, however hes at his maximum velocity, so...
8m/s(man) - 4.5m/s(train) = 3.5 m/s more

(1.111m) / (3.5m/s) = .317seconds more to reach the train

So if it takes 2.222 seconds to approach the train at 8.888m, it should take
2.222 + .317 =2.529 seconds to reach the train completely

Last but not least is to figure out the total distance traveled in that time frame:

(Trains velocity) * (total time)

(4.5m/s)*(2.529s)=11.3805m

If you really have nothing else but time, then you can't. There must be some other shred of information. Search around. Look under rocks.

Fine.  You just need a conversion at some point between seconds and hours.
Let's do it this way:

First, let's review one very useful definition of acceleration:

                   Acceleration = (change in speed) / (time for the change)

The question is asking for a time, so let's massage this equation to get
time all by itself.
Multiply each side by (time for the change) :

                   (Acceleration) x (time for the change) = (change in speed)

Divide each side by (acceleration) :

                   Time for the change = (change in speed) / (acceleration).

There it is.  The problem gives you the change in speed and the acceleration.

                    Change in speed = (110 - 80) = 30 km/hr

                    Acceleration = 1.6 m/s²

Here's where the only difficulty shows up.  One is in hours and the
other one is in seconds.
The way you change the units of a quantity is to multiply it by
a fraction whose value is ' 1 '.  That way you don't change the value
of the original quantity.

How does that help ?  You just pick the fraction very carefully !  Remember
that the value of a fraction is ' 1 ' if the numerator and denominator are both
the same value.  Here are some fractions that are equal to ' 1 ' :

    12 inches/1 foot     60 seconds/1 minute     1 meter/100 centimeters

See how that works ?  Multiply one of these by a quantity, and it changes
the units of the quantity, but not its value.

Let's change  1.6 m/s²  to m/hr² .

1 hour = 3,600 seconds  so the fraction is  (3600 sec/hr) .

The unit conversion is:  (1.6 m/s²) x (3600 sec/hr) x (3600 sec/hr)

I multiplied by the fraction twice because we need sec² on top to cancel
the s² on the bottom.

Multiply this out and you get  20,736,000 m/hr². 
Don't let the big number scare you.

Now go back to our equation for the time:

                    Time for the change = (change in speed) / (acceleration)

Change in speed = 30 km/hr

Acceleration was 1.6 m/sec²  but we found that
that's the same as 20,736,000  m/hr² .
And also, since the change in speed is in km,
let's change this one to  20,736 km/hr² .

                     Time for the change = (30 km/hr) / (20,736 km/hr²)

                     30 / 20,736 = 0.0001447 hour

                                      =  5.2 seconds.

I realize that this has turned into a nightmare.  I probably lost you, and
the answer is likely wrong because the numbers got so ugly.  I apologize. 
The reason for all the ugly numbers is that I apparently picked the wrong
number to change the units.  It probably would have been a lot neater to
convert the change in speed to  m/sec  instead of changing the acceleration
to  m/hr² .  But the principle is the same.  Make a fraction out of the unit you
have and the new unit you want, wit the same value on top and bottom, and
use it to convert the units of one number to match the units of the others.
========================================

Change in speed = (30 km/hr) x (1000 m/km) x (1 hr / 3600 sec) = 8.333 m/sec

Time = (change in speed) / (acceleration) = (8.333 m/s) / (1.6 m/s²) = 5.2 seconds

Yes, this way was certainly better and prettier.
But at least I got the same answer going the long tough way!

I think one popular definition of acceleration might help you here:

Acceleration = (change in speed) / (time for the change)

Multiply each side of this equation by (time for the change) :

Change in speed = (acceleration) x (time for the change)

The problem gives you the acceleration and the time, so

Change in speed = (0.73 m/s²) x (7.1 s)  =  5.183 m/s

Before the acceleration, the car's speed was 7.6 m/s . 
Then the car accelerated for a while. 
During the acceleration, its speed grew by 5.183 m/s.
The final speed is  (7.6 + 5.183) =  12.783 m/s  .

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At the speed of 340 m/s , sound travels (1.10 x 340) = 374 meters in 1.10 s .

That's the round-trip distance from my mouth to the other side of the canyon
and back, so the other side of the canyon is  374/2 = 187 meters away. 
(about 614 feet)