![]() ![]() What if the piano bounced back up during the collision? Since momentum is a vector, a piano going down and then up would have a much larger change in momentum than one that just stops. It increases the time over which it stops you and decreases the force.īut what if the piano didn't stop? What if the piano kept falling down as it crashed through the roof? In this case, the piano would have a smaller change in momentum and a smaller impact force. This is essentially what an air bag in your car does. A large increase in collision time can mean decrease in impact force. Now if I put in different impact times, I get the following plot. Just for fun, what if the collision time was a little longer or a little shorter? All I need to do is to rearrange the momentum principle equation above to solve for the force the roof pushes on the piano instead of the time. If I convert these values and do exactly the same thing as before, what value do I get for impact time? First, I get an impact velocity of 21.6 m/s and second I get a collision time of 0.107 seconds (or 0.1 seconds). They estimated an impact force of 55,000 pounds. Ok, what about another example? Later, the MythBusters dropped a 2,600 pound piano (filled with sand) from a height of 75 feet. Putting in the values for the knowns, I get a time interval of 0.109 seconds. Now, I know everything in this expression except for Δt. Just to be clear: the final velocity is zero and the initial velocity is in the negative y-direction. Since this all happens just in the vertical direction, I can write this as the scalar equation: The force the roof pushes up is the same force that the piano pushes on the roof - this is the impact force. There are two forces acting on the piano: gravity and the roof pushing up. Let me start with a force diagram showing the forces acting on the piano during the collision. Since we know the estimated impact force from the show (12,000 pounds = 53,379 Newtons), the impact time can be calculated. I also know the final momentum since I can assume it comes to a rest. I know the starting momentum (from the velocity right before it collides). This gives a relationship between the net force on an object and that object's change in momentum. So as this piano collides with the roof, how could you estimate this impact force? Let's start with the momentum principle. Oh, here's tip - just type into google: "17.28 m/s in mph" and you will get the conversion. That's essentially the answer they said on the show. Convert this to mph and you get 38.7 mph. Now put in a height of 15.24 meters and value for g of 9.8 m/s 2 and you get a final speed of 17.28 m/s. Now I can put this together and solve for the final velocity. Since the piano starts from rest, the initial kinetic energy is zero. This work would be equal to the change in kinetic energy. Since the gravitational force is pulling in the same direction as the piano is moving, θ would be zero. That would be like having your cake and eating it too. In that case, there would be a gravitational potential energy but no work done by gravity. ![]() But should there be a gravitational potential energy? Yes, you could do it that way if you chose the Piano and the Earth as the system. Really, there is only one force on the piano - the gravitational force. In order to use the work energy, I need to find what forces do work on the piano as it falls. ![]()
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