Is anyone else struggling with Applications of Calculus?? Does anyone have any tips on how to do those describe the motion of a particle questions??
I'll just go through what I know from SACE, it may be a little different for you, but I'm sure that the fundamentals are still there.
You'll always be given the function for s, displacement, at the stage of the course. This tells you the displacement (how far away) the particle is from the origin. When:
s(t) = 0, the particle is at the origin
s(t) < 0, the particle is to the left of the origin
s(t) > 0, the particle is to the right of the origin
Hopefully you know that the derivative of the displacement function results in the velocity function. This is due to the velocity being how quickly the particle is moving / how quickly it's displacement is changing. Hence:
s'(t) = v(t)
When:
v(t) = 0, the particle is at rest/not moving
v(t) < 0, the particle is moving to the left
v(t) > 0, the particle is moving to the right
And the rate of which the velocity changes, is the acceleration, as if I am accelerating in the same direction as I am travelling, my velocity will also increase at said rate. Therefore acceleration is how quickly the velocity of the particle is increasing/decreasing.
v'(t) = a(t)
a(t) = 0, velocity may be at a maximum or minimum
Now, the most fundamental concept in one of these questions is determining :
Another key concept is the be able to determine using sign tests, when the speed is increasing or decreasing.
If the signs of v(t) and a(t) are the same, then the speed of particle P is increasing
If the signs of v(t) and a(t) are different, then the speed of P is decreasing.
This is due to the fact that if I am travelling to the left, but I am being pushed to the right, my speed will decrease as I won't be able to walk to the left for much longer.