# A hammer with 10 J of kinetic energy hits a nail and pushes it 5.0 mm into a plank.

Both the hammer and nail come to rest after the collision.
What is the average force that acts on the nail while it moves the 5.0 mm?

## KE=mv²/2

v²= 2•KE/m
a= v²/2s
F=ma=mv²/2s= 2m•KE/2s m=
= KE/s=10/0.005 =2000 N

## To find the average force acting on the nail, we can use the work-energy principle. The work done on the nail is equal to the change in its kinetic energy. We can calculate the work done using the formula:

Work = force x distance

Since the nail is pushed into the plank, the force acting on the nail is opposite to its direction of motion. Therefore, we can rewrite the formula as:

Work = - force x distance

Given that the nail is pushed 5.0 mm into the plank, we can convert this distance to meters by dividing it by 1000:

Distance = 5.0 mm ÷ 1000 = 0.005 m

We also know that the kinetic energy of the hammer is 10 J and it comes to rest after the collision. This means that the entire kinetic energy of the hammer is transferred to the nail.

Therefore, the work done on the nail is equal to the change in its kinetic energy:

Work = Change in Kinetic Energy

Since the nail comes to rest, its final kinetic energy is zero. Therefore, the change in kinetic energy is:

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
= 0 J - 10 J
= -10 J

Now, we can use the work-energy principle to find the average force:

-10 J = - force x 0.005 m

Rearranging the equation, we can isolate the force:

force = -10 J ÷ 0.005 m
= -2000 N

The average force that acts on the nail while it moves the 5.0 mm into the plank is -2000 N. Note that the negative sign indicates that the force is acting in the opposite direction of the nail's motion.

## To find the average force that acts on the nail while it moves 5.0 mm, we can use the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy. So, in this case, the work done on the nail is equal to the change in its kinetic energy.

The work done on an object can be calculated using the formula:

Work = Force x Distance x cos(θ)

Where:
- Force is the average force
- Distance is the displacement of the object (in this case, 5.0 mm or 0.005 meters)
- θ is the angle between the force and displacement vectors (which is assumed to be zero since the force and displacement are in the same direction)

Since the nail comes to rest after the collision, its final kinetic energy is zero. The initial kinetic energy of the hammer is given as 10 J. The change in kinetic energy is then:

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
= 0 J - 10 J
= -10 J

Since the work done on the nail is equal to the change in its kinetic energy, we have:

Work = -10 J

Using the formula for work, we can write:

-10 J = Force x 0.005 meters x cos(0)

Since cos(0) = 1, the equation simplifies to:

-10 J = Force x 0.005 meters

Now, we can solve for the average force:

Force = -10 J / 0.005 meters
= -2000 N

The negative sign indicates that the force is acting in the opposite direction to the displacement (decelerating the nail). Therefore, the average force that acts on the nail while it moves 5.0 mm is 2000 N in the direction opposite to the motion.