FIGURE EX10.25 is the potential-energy diagram for a 20 g particl...

Question

FIGURE EX10.25 is the potential-energy diagram for a 20 g particle that is released from rest at x = 1.0m. (b) What is the particle's maximum speed? At what position does it have this speed?

Accepted Answer

Hey everyone. So this problem is dealing with potential energy graphs. Let's see what it's asking us. Suppose a small object weighing only eight g is placed at a position X equals two m. On a potential energy diagram. The object is initially at rest. And the figure below shows how the object's potential energy changes as it moves from its initial position. Calculate the object's maximum velocity and determine the position where that maximum velocity is reached our potential or our answer choices here R A maximum velocity is equal to 17.3 m per second at X equals four m. B maximum velocity is equal to 14.2 2nd, 14.2 m per second at X equals six m. C maximum velocity is equal to 27.4 m per second at X equals eight m or D. Maximum velocity is equal to 22.8 m per second at X equals two m. OK. So the first thing we can do here is recall our conservation and energy equation. It is given by delta E equals zero, right? So there is no change in energy from our final energy to our initial energy or E F is equal to E I. In turn, we can recall that our total energy is our kinetic energy plus our potential energy. And our kinetic energy is given by one half M V squared. So if we break this first equation out, we have K I plus U I equals K F plus U. And our initial, our object is initially at rest, which means our initial kinetic energy is zero, We can look at the graph. So the object is initially placed at X equals two m. So at X equals two m, our potential energy is five joules. So we know from our conservation energy equation that at any point, our ener total energy is equal to a kinetic energy plus our potential energy. If we have no kinetic energy at this point at two years, then our only energy is our potential energy which is five jewels. So at any other point on the graph again, because of the conservation of kinetic energy, our total energy is equal, sorry or co sorry because of the conservation of energy, our total energy is equal to five Jes. We can see by the relationship between kinetic energy, velocity and potential energy that our maximum velocity happens when our potential energy is at a minimum. So the graph has a minimum potential energy at X equals eight m. That is the bottom valley of this graph. And at that point, our potential energy is equal to two jolt. So when we have maximum velocity, our energy again is equal to kinetic energy plus potential energy. So our total energy is five jewels is equal to our kinetic energy plus the potential energy of two jewels. So our kinetic energy is equal to three Jews. Now we can plug that in to our kinetic energy equation to find our maximum velocity. So K equals one half and B squared. So our maximum velocity is equal to two K divided by M take a square rid of that. And so we plug in our known values here, you get two multiplied by K which is three Jews all divided by R mass and the mask. And the problem was given as eight g. So we're gonna keep that in right that over here, right? And keep that in standard units. So eight g can also be written as eight times 10 to the negative three kg. So eight times 10 to the negative three kg is our mass. And then we plug that into our calculator and we get 27. m per second. And so that is the answer to this problem. You have a speed of 27.4 m per second at X equals eight m. And so that aligns with answer choice C. That's all we have for this one. We'll see you in the next video.

FIGURE EX10.25 is the potential-energy diagram for a 20 g particle that is released from rest at x = 1.0m. (b) What is the particle's maximum speed? At what position does it have this speed?

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