A builder is hammering nails into a wall. The hammer has a mass of 3.3 kilograms, and each nail has a mass of 308 grams. Given that the hammer hits each stationary nail at a speed of 8.2 metres per second, use the principle of conservation of momentum to find the speed of the hammer and nail directly after the impact.
Well, this question has told us exactly what we’re going to need to do to answer it. We’re going to use the principle of conservation of momentum. This says that the total momentum before a collision or impact is equal to the total momentum after that collision. And the formula for calculating the momentum of an object is momentum equals mass times velocity. So, let’s look carefully at this scenario.
Before the collision, we have two separate objects. We have a hammer which has a mass of 3.3 kilograms, and it moves at a speed of 8.2 metres per second. We also have a nail. Now, the measurements for the nail are in grams, so let’s divide by 1000. And we see that each nail has a mass of 0.308 kilograms. We’re told that each nail is stationary; that is, they’re moving at zero metres per second.
After the collision, the hammer and nail move as one object. Their combined mass will be 3.3 kilograms plus 0.308 kilograms. That’s 3.608 kilograms. Now, at this stage, it’s important to realise that we could have represented each of our masses in grams. It really doesn’t matter at this stage as long as we’re consistent.
We’re looking to calculate the speed of the hammer and nail directly after the impact. So, let’s call the speed after 𝑣 metres per second. Now, let’s consider the total momentum before the collision. We know that the hammer is moving at 8.2 metres per second, so mass times velocity is 3.3 times 8.2. The nail is stationary, so its speed is zero metres per second, and its momentum is 0.308 times zero.
We know that the total momentum before the collision is equal to the total momentum immediately after the collision. Well, the momentum of the hammer and nail combined is its mass times its velocity. So, that’s 3.608 times 𝑣. Let’s simplify a little. 3.3 times 8.2 is 27.06. And of course, 0.308 times zero is zero. It does make a lot of sense that the momentum of the nail before the impact was zero. So, our equation is 27.06 equals 3.608𝑣.
We solve this equation for 𝑣 by dividing both sides by 3.608. And 27.06 divided by 3.608 is 7.5. We, of course, must be consistent with our units. We have measured in metres per second throughout. So, the velocity of the object after the collision is 7.5 metres per second. Now, in fact, we were really only interested in the speed, so that’s the magnitude of velocity. In other words, the direction doesn’t matter. So, we can see that the speed of the object is 7.5 metres per second and that it’s moving in the same direction as the hammer, as we would expect.