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A spear from a spear gun is fired horizontally underwater.
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The spear’s mass is 250 grams and it is fired with an initial horizontal velocity of 16 metres per second.
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The spear travels a distance of 10 metres horizontally before its horizontal speed becomes zero.
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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?
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Okay, so in this question, we’ve been told that we’ve got a spear being fired from a speargun.
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So let’s say that this is our spear, which we’ve just fired from a very badly drawn speargun.
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We’ve been told the spear’s initial velocity, which happens to be 16 metres per second.
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And we’re told that the spear travels a distance of 10 metres horizontally before its horizontal speed becomes zero metres per second.
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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.
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And we’ve been asked to work out this horizontal average force in the direction of its motion.
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Now, before we do any working out, we know that the spear is moving as we’ve drawn it to the right.
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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.
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In other words, the water is going to have to exert a force to the left.
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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.
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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.
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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.
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Now, in this case, our object of course is the spear.
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And we already know the mass of the spear.
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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.
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So we can use the suvat equation: 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠.
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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.
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So let’s rearrange and plug those values in.
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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.
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This way, on the right-hand side, the twos cancel and the 𝑠s also cancel.
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We’re just left with the acceleration on the right.
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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.
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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.
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Now, at this point, we know all of the quantities on the right-hand side of the equation.
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So we can just plug them in.
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But before we do, we need to check that they’re all in their standard units.
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So first of all, the initial velocity is in metres per second which is the standard unit of velocity.
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The same is true for the final velocity; it’s also in metres per second.
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And the distance travelled in a straight line has been given in metres, which is the standard unit of distance.
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However, the mass of the spear has been given in grams, not kilograms.
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So we need to convert 250 grams to kilograms.
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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.
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And so we find that 250 grams is equal to 0.25 kilograms.
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So let’s write that down over here and let’s now look at plugging in all the values into our equation here.
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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.
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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.
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And the standard unit of force is the newton.
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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.
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And we’ve already discussed why the force is going to be negative.
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So this answer makes sense.
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Therefore, we found our final answer.
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The horizontal average force exerted by the water on the spear in the direction of its motion is negative 3.2 newtons.