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Mechanics M2 - Variable Acceleration (Vectors)

I need help with an M2 question on variable acceleration (vectors). Here is the question: At time t seconds (t》0), a particle P is moving in a plane with acceleration a=(5t-3)i+(8-t)j. When t=0, the velocity of P is (2i-5j)ms^-1. Find the value of t for which P is moving parallel to i-j. Could someone help me out please?
First, you're given some useful information about the velocity, so you might want to start off by finding v, noting that

a=dvdt\displaystyle a=\dfrac{dv}{dt}

and not forgetting the constant of integration (a vector which you can then find using the info).

Let me know when you've done that :smile:
(edited 10 years ago)
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Original post by Indeterminate
First, you're given some useful information about the velocity, so you might want to start off by finding v, noting that

a=dvdt\displaystyle a=\dfrac{dv}{dt}

and not forgetting the constant of integration (a vector which you can then find using the info).

Let me know when you've done that :smile:


Hey there, I did that and got: v= (2.5t^2-3t+2)i+(8t-0.5t^2-5)j ms^-1. I'm not sure what to do next though.
Original post by 123456789012
Hey there, I did that and got: v= (2.5t^2-3t+2)i+(8t-0.5t^2-5)j ms^-1. I'm not sure what to do next though.


The key now is to think of what it means for it to be parallel to i-j. The direction i-j i.e (1,−1)(1,-1) is where the j component is the negative of the i component. Use this to solve for t :smile:
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Original post by Indeterminate
The key now is to think of what it means for it to be parallel to i-j. The direction i-j i.e (1,−1)(1,-1) is where the j component is the negative of the i component. Use this to solve for t :smile:


That's exactly what I don't know! I would know what to do if the question told me the particle was moving parallel to i or j only. However, I'm unsure on how to go about doing this question.
Original post by 123456789012
That's exactly what I don't know! I would know what to do if the question told me the particle was moving parallel to i or j only. However, I'm unsure on how to go about doing this question.


The word 'parallel' means that it's travelling in the same direction as the vector i-j at some time, t.

Note the characteristic of i-j is that the j component is the negative of the i component, which defines the direction of it.

So, you must set up an equation so that the vector v conforms to this definition.

Can you do it now? :smile:
Original post by 123456789012
I need help with an M2 question on variable acceleration (vectors). Here is the question: At time t seconds (t》0), a particle P is moving in a plane with acceleration a=(5t-3)i+(8-t)j. When t=0, the velocity of P is (2i-5j)ms^-1. Find the value of t for which P is moving parallel to i-j. Could someone help me out please?


Integrate the acceleration to get the velocity then put equal to K(i-j) where k is a constant.
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Original post by brianeverit
Integrate the acceleration to get the velocity then put equal to K(i-j) where k is a constant.


Thanks a lot! That makes sense! You've helped me out a lot. :biggrin:

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