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Graphs of the $ position $ functions of two particles are shown, where $ t $ is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.

a) Particle is speeding up on $(1,2)$ and $(3,4)$

Particle is slowing up on $(0,1)$ and $(2,3)$

b) speeding up: $(1,2),(3,4) \quad$ slowing down: $(0,1),(2,3)$

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

here we have the graph of a position function and we want to know when the particle is speeding up and when it's slowing down, it'll be speeding up when the velocity and acceleration have the same sign, either both positive or both negative. And it will be slowing down when they have different signs one positive and one negative. So what we're going to do is break our time, intervals down and analyze what's happening with the velocity and acceleration in each interval. So we have 0 to 1 1 to 2, 2 to 3 and 3 to 4. Something is changing for each interval. So now we're going to find the sign of the velocity and the sign of the acceleration. When the position graph is increasing, the velocity is positive. So we see it increasing from 0 to 1, and we see it increasing from 3 to 4. When the position equation is decreasing, the velocity is negative, so we see a decreasing from 1 to 3. Now, the acceleration is the second derivative of position, and the second derivative relates to the con cavity of the curve con que but versus conch it down. If the curve is conquered down. The second derivative will be negative. And if the curve is Khan gave up, the second derivative will be positive. So we see that this curve is conclave, down from 0 to 2. So even negative acceleration. And then it's Khan gave up from 2 to 4, so we have a positive acceleration. Now let's look and see when both of these signs are positive. Velocity and acceleration are both positive from 0 to 4, and they're both negative from 1 to 2. So that's an indication of speeding up. So we have speeding up from times 1 to 2 and from times 3 to 4. Now, look for when they have opposite signs from 0 to 1 and from 2 to 3. So that means it's slowing down 0 to 1, 2 to 3. Okay, let's do something similar for the second graph. So again, we're going to be analyzing the velocity and the acceleration to figure out whether they're positive or negative. So we have the interval from 0 to 1 from 1 to 2 from 2 to 3, and from 3 to 4, we have the velocity and we have the acceleration. Okay, So this graph is decreasing from 0 to 3, and so it has a negative velocity from 0 to 3. This graph is increasing from 3 to 4, so it has a positive velocity from 3 to 4. This graph is conclave, up from 0 to 1, so it has a positive acceleration. Then it's Con Cape, down from 1 to 2, so it has a negative acceleration. Then it's called Cave up from 2 to 4, so it has a positive acceleration again. Now let's look for where the signs are the same. The signs are the same from 1 to 2 and 3 to 4, so that would be when it's speeding up. The signs are different from 0 to 1 and 2 to 3. That would be when it's slowing down.