### Video Transcript

A bullet has a mass of 2.60
grams and moves horizontally at a speed of 335 meters per second as it collides
with a stack of eight pine boards, each 0.750 inches thick. The bullet decelerates as it
penetrates the boards. And the bullet comes to rest
just as it has moved the full distance through all eight boards. Find the average force exerted
by the boards on the bullet. Assume that the motion of the
boards due to the bullet’s impact is negligible.

Looking to solve here for the
average force exerted by the boards on a bullet, we can call that force 𝐹. And we’ll start on our solution
by drawing a diagram of this situation. We have in this scenario a
bullet with an initial speed 𝑣 sub 𝑖 of 335 meters per second encountering a
stack of eight pine boards. Each of the boards has a
thickness 𝑇 of 0.750 inches. And we’re told that by the time
the bullet makes it all the way through the last board, it comes to a stop.

Knowing all this, we want to
solve for the average force 𝐹 that the boards exert on the bullet as it
decelerates. Seeking to solve for average
force may remind us of Newton’s second law of motion, which says that the net
force on an object equals its mass times its acceleration 𝑎. Using this law, we want to
solve for 𝐹. And we’re given 𝑚 in the
problem statement. But we don’t yet know the
acceleration of the bullet as it comes to a stop. However, if we assume that
acceleration 𝑎 is constant, then that means the kinematic equations apply for
describing the motion of the decelerating bullet.

Looking over these four
equations of motion, we see that the second one helps us solve for what we want
to know — acceleration — in terms of values we’re given. Written in terms of the
variables for our particular situation, we can say that the final speed of the
bullet squared is equal to its initial speed squared plus two times its
acceleration times the distance it travels eight times the thickness of a
board.

We know the bullet ends up at
rest. So 𝑣 sub 𝑓 is equal to
zero. So when we rearrange to solve
for 𝑎, it equals negative 𝑣 sub 𝑖 squared all over two times eight 𝑇. We can substitute this
expression for acceleration in for 𝑎 in our equation for force. And we understand that even
though the minus sign implies that the bullet is decelerating, which is
true. Since we want to solve for the
average force the boards exert on the bullet, which will be positive, we’ll
change that to a plus sign.

Looking at this expression, we
were given the mass of the bullet 𝑚 in the problem statement as well as the
initial speed of the bullet 𝑣 sub 𝑖. We’re also told the thickness
𝑇 of the boards. But that thickness is currently
expressed in inches and we like to convert it to meters. 0.750 inches is approximately
1.905 centimeters. When we plug our values into
this expression, we’re careful to use a mass in units of kilograms and a
distance of the thickness of each board in units of meters. Entering this expression on our
calculator, it comes out to 960 newtons. To two significant figures,
that’s the average force that the boards exert on the bullet.