A spear from a spear gun is fired
horizontally underwater. The spear’s mass is 250 grams and
it is fired with an initial horizontal velocity of 16 metres per second. The spear travels a distance of 10
metres horizontally before its horizontal speed becomes zero. What horizontal average force does
the water apply on the spear in the direction of its motion before the spear’s
horizontal velocity becomes zero?
Okay, so in this question, we’ve
been told that we’ve got a spear being fired from a speargun. So let’s say that this is our
spear, which we’ve just fired from a very badly drawn speargun. We’ve been told the spear’s initial
velocity, which happens to be 16 metres per second. And we’re told that the spear
travels a distance of 10 metres horizontally before its horizontal speed becomes
zero metres per second.
What we’ve been asked to do is to
work out the horizontal average force applied by the water, which in this case
surrounds the spear. And we’ve been asked to work out
this horizontal average force in the direction of its motion. Now, before we do any working out,
we know that the spear is moving as we’ve drawn it to the right. And if the water is going to slow
down and eventually stop the spear, then the force that it exerts is going to have
to be in the opposite direction to the spear’s motion.
In other words, the water is going
to have to exert a force to the left. This is the only way for the spear
to slow down because if the force was exerted in the same direction as the spear’s
motion, then the spear would accelerate even further. And therefore, the horizontal
average force in the direction of the motion of the spear is going to be negative
because it’s actually not acting in the direction of the motion, but it is acting in
the opposite direction, hence negative.
Now, to work out the size or
magnitude of the horizontal average force exerted by the water, we can recall
Newton’s second law of motion, which says that the net or resultant force on an
object is equal to the mass of that object multiplied by the acceleration
experienced by the object.
Now, in this case, our object of
course is the spear. And we already know the mass of the
spear. And we can work out the average
acceleration experienced by the spear based on the information that we have in the
diagram because the information that we have is the initial velocity of the spear
which we’ll call 𝑢 which is 16 metres per second, we also have the final velocity
of the spear which we’ll call 𝑣, which happens to be zero metres per second, and we
have the distance travelled by the spear in a straight line which we’ll call 𝑠 and
this happens to be 10 metres.
So we can use the suvat equation:
𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠. And we can rearrange this to give
us the value of the acceleration experienced by the spear because we have all the
other quantities in the equation: we’ve got 𝑣, we’ve got 𝑢, and we’ve got 𝑠. So let’s rearrange and plug those
First, we can subtract 𝑢 squared
from both sides so that 𝑢 squared on the right cancels out and then we divide both
sides of the equation by two 𝑠. This way, on the right-hand side,
the twos cancel and the 𝑠s also cancel. We’re just left with the
acceleration on the right.
Now, at this point, we could plug
in our values to work out the value of 𝑎 or we could substitute this into our
equation 𝐹 is equal to 𝑚𝑎. When we do this, we find that the
force exerted on the spear is equal to the mass of the spear multiplied by the final
velocity of the spear squared minus the initial velocity of the spear squared
divided by two times the distance travelled by the spear in a straight line.
Now, at this point, we know all of
the quantities on the right-hand side of the equation. So we can just plug them in. But before we do, we need to check
that they’re all in their standard units.
So first of all, the initial
velocity is in metres per second which is the standard unit of velocity. The same is true for the final
velocity; it’s also in metres per second. And the distance travelled in a
straight line has been given in metres, which is the standard unit of distance. However, the mass of the spear has
been given in grams, not kilograms. So we need to convert 250 grams to
To do this, we can recall that one
gram is equal to one thousandth of a kilogram, at which point we can multiply both
sides of the equation by 250. And so we find that 250 grams is
equal to 0.25 kilograms. So let’s write that down over here
and let’s now look at plugging in all the values into our equation here.
When we do this, we find that the
force is equal to 0.25, that’s the mass, multiplied by zero squared, final velocity
squared, minus 16 squared, initial velocity squared, divided by two times 10, that’s
two times the distance travelled in a straight line.
And because we’ve been careful to
convert everything on the right-hand side to standard units, then the left-hand side
is going to be in standard units as well. And the standard unit of force is
the newton. So when we evaluate the right-hand
side of the equation, we find that the force exerted on the spear is negative 3.2
newtons. And we’ve already discussed why the
force is going to be negative. So this answer makes sense.
Therefore, we found our final
answer. The horizontal average force
exerted by the water on the spear in the direction of its motion is negative 3.2