Relationship between momentum and kinetic energy

ke = 1/2 * m * V^2            kinetic energy = 1/2 * mass * velocity squared
ke = 1/2 * m * v * v           kinetic energy = 1/2 * mass * velocity * velocity
p = m * v                            momentum = mass * velocity
ke = (m * v) * (1/2 * v)      kinetic energy = momentum * 1/2 * velocity
An object's kinetic energy due to its motion is equal to the product of its momentum
and 1/2 its velocity.    ke = p * (1/2 * v)
                            also
An object's momentum is the quotient of the division of its kinetic energy
by 1/2 of its velocity.    p = ke / (1/2 * v)

The collision does not take 4 sec; I am pointing out that the quantity of Ft to make the initial motion is equal to the quantity of force * time needed to produce the final motion.

You mention "moment of inertia" is this moving in a circle?

@ Delburt Phend

? 1.  What is the momentum of 3 Newtons of force    times   25 seconds of time ?
? 2. What is the difference between an apple ?

The apple will have 75 kg m/sec of momentum.

                Relationship between momentum and kinetic energy

ke = 1/2 * m * V^2            kinetic energy = 1/2 * mass * velocity squared
ke = 1/2 * m * v * v           kinetic energy = 1/2 * mass * velocity * velocity
p = m * v                            momentum = mass * velocity
ke = (m * v) * (1/2 * v)      kinetic energy = momentum * 1/2 * velocity

An object's kinetic energy due to its motion is equal to the product of its momentum
and 1/2 its velocity.    ke = p * (1/2 * v)
                  also
An object's momentum is the quotient of the division of its kinetic energy
by 1/2 of its velocity.   p = ke / (1/2 * v)
... ... ... ... ... ... ...
... ... ... ... ... ... ...
                                  scenario 0

force = 3 N
time  = 25 seconds
mass = zero

force  = mass times acceleration
momentum = mass times velocity

momentum = zero
... ... ... ... ... ... ...
                                  scenario 1
force = 3 N
time = 25 seconds
mass = 15 kg

force  = mass times acceleration
momentum = mass times velocity

momentum = 15 kg * 5m/s ..  "75 momentum"

kinetic energy = 1/2 * m * v^2

velocity = 5 m/s
     because
1 Newtons of force will cause 1 kg to accelerate to 1 meter / second in 1
second  (by definition)
     and
1 Newton of force will cause 15 kg to accelerate to 1/15 (0.0666666666666666
6666666666666667)  meter / second in 1 second
     and
3 Newtons of force will cause 15 kg to accelerate to ( 0.0666... times 3) which is
0.2 meter/second in 1 second
    and
3 Newtons of force will cause 15 kg to accelerate to, 0.2  times 25, in 25 seconds
    and
this is equal to 5 meters per second velocity (0.2 * 25 = 5)

A 15 kg mass moving at 5 m/s has 187.5 joules of  kinetic energy
                            Ek = 1/2 * Mass * Velocity ^2
kinetic energy = 1/2 * 15 kg * 5m/s^2 ... 1/2 * 15 kg * 25 =  187.5 joules

5 m/s ^2 = 25
15 kg * 25 = 375
375 / 2 = 187.5 joules

An object's momentum is the quotient of the division of its kinetic energy
by 1/2 of its velocity.    p = ke / (1/2 * v)
187.5 joules / 1/2 * 5 m/s   as in  187.5 joules / 2.5 m/s  as in  "75 momentum"

momentum = mass * velocity
15 kg of mass * 5 m/s =                                                             "75 momentum"

Momentum = force * time 
3 Newtons  * 25 seconds =                                                          "75 momentum"
... ... ... ... ... ... ...
... ... ... ... ... ... ...
                                  scenario 2

force = 3 N
time  = 25 seconds
mass = 4 kg

force  = mass times acceleration
momentum = mass times velocity

momentum = zero
... ... ... ... ... ... ...
force = 3 N
time = 25 seconds
mass = 4 kg

momentum = 4 kg * 0.75 m/s = 3

kinetic energy = 1/2 * m * v^2

velocity =  18.75  m/s
     because
1 Newtons of force will cause 1 kg to accelerate to 1 meter / second in 1
second  (by definition)
     and
1 Newton of force will cause 4 kg to accelerate to 1/4 (0.25)  meter / second in 1 second
     and
3 Newtons of force will cause 4 kg to accelerate to 0.25 times 3 which is
0.75 meter/second in 1 second
    and
3 Newtons of force will cause 4 kg to accelerate to 0.75  times 25, in 25 seconds
    and
this is equal to 18.75 meters per second velocity (0.75 * 25 = 18.75)
... ... ... ... ...
A 4 kg mass moving at 18.75 m/s has  ?  joules of  kinetic energy
                            Ek = 1/2 * Mass * Velocity ^2
kinetic energy = 1/2 * 4 kg * 18.75 m/s^2 ... 1/2 * 4 kg * 351.5625 = 703.125 joules

18.75 m/s ^2 = 351.5625
4 kg * 351.5625 = 1406.25
1406.25 / 2 = 703.125 joules

An object's momentum is the quotient of the division of its kinetic energy
by 1/2 of its velocity.    p = ke / (1/2 * v)
703.125 joules / 1/2 * 18.75 m/s  or stated as 703.125 joules / 9.375 m/s or stated
as in                                                "75 momentum"

momentum = mass * velocity
4 kg of mass * 18.75 m/s =           "75 momentum"

Momentum = force * time
3 Newtons  * 25 seconds =           "75 momentum"

                       Momentum = force * time

                                       cool !