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Need help with M2, ex 1C

So i am really stuck with these two questions:

6) A particle P is moving along the x-axis with velocity v=4t-2t^2. When t=0, P is at x=3. Find a) the position of P when t=2 b)the maximum velocity attainted by P c) the distance OP when the velocity is maximum

7)A particle P os moving along the x-axis. When t=0, the velocity of P is 4.5 m/s. At time ts the acceleration of P is given by (3t-6) m/s^2. Find the time when the particle returns to its starting point.

For the 6th question, i did the first two parts but i think my method is wrong so will really appreciate it if someone does it in detail.
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Avatar for nuodai
nuodai
17
6a) You should know that
Unparseable latex formula:

x = \displaystyle \int v\ \mbox{d}t

, and you can find the constant of integration given the initial conditions (so sub in x=3 and t=0 to find the constant of integration, and then t=2 to find the value of x)
6b) You can find the maximum of a curve by differentiating... so do that
6c) You can use the answers to part a and b to answer this

7) Again you need to integrate and use the initial conditions to find something in the form x=f(t)x = f(t), and then sub in x=0 and solve to find the time.
Start with the definition of velocity:

v= dx/dt

To get position from this, need to integrate with respect to time, i.e

x = integral(4t-2t^2)dt

Gives us

x = 2t^2-2/3t^3 + constant

the info in question says t=0, x = 3 so can see that constant = 3, giving

x = 2t^2-2/3t^3 + 3

as equation for position. From there it is putting in numbers for part a.

for part b, remember that a maximum (or minimum) is found by setting the differential of something to 0. In this case dv/dt = 0 would be a good bet!

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