# Thread: Mechanics Problem - Vectors

1. ## Mechanics Problem - Vectors

A particle has position vector 2i + j initially and is moving with speed 10m/s in the direction 3i - 4j. Find its position vector when t = 3 and the distance it has travelled in those 3 seconds.

My solution:
Position vector = initial position + displacement
since displacement = velocity x time
then position vector= (2i + j) + 3(3i - 4j)
= 2i + j + 9i - 12i
= 11i - 11j
and distance is straightforward, distance = speed x time so 10 x 3 which is 30m.
My solution is however wrong, becuase in the question it says (3i - 4j) is the direction the particle is travelling in, not the velocity, so im kinda stuck and the right answer should be 20i - 23j according to the back of the book.

How do i do this question? Also, if u find the velocity vector, and work out the magnitude, you should get the speed right? So if initial velocity is (2i + 3j) then the magnitude of this vector should equal the speed which is 10 m/s but it does not, the magnitude of this vector is the sqrt (2^2 + 3^2) which is root 13 and not 10 m/s so how does that work? Thanks for you help.

2. Originally Posted by MingMing
A particle has position vector 2i + j initially and is moving with speed 10m/s in the direction 3i - 4j. Find its position vector when t = 3 and the distance it has travelled in those 3 seconds.

My solution:
Position vector = initial position + displacement
since displacement = velocity x time
then position vector= (2i + j) + 3(3i - 4j)
= 2i + j + 9i - 12i
= 11i - 11j
and distance is straightforward, distance = speed x time so 10 x 3 which is 30m.
My solution is however wrong, becuase in the question it says (3i - 4j) is the direction the particle is travelling in, not the velocity, so im kinda stuck and the right answer should be 20i - 23j according to the back of the book.

How do i do this question? Also, if u find the velocity vector, and work out the magnitude, you should get the speed right? So if initial velocity is (2i + 3j) then the magnitude of this vector should equal the speed which is 10 m/s but it does not, the magnitude of this vector is the sqrt (2^2 + 3^2) which is root 13 and not 10 m/s so how does that work? Thanks for you help.
$\displaystyle \displaystyle \vec{s} = \vec{s_0} + \vec{v}t$

Note that 3i - 4j is a vector in the direction of the velocity. It is not the velocity. The velocity will be the unit vector in the direction of 3i - 4j times the speed. Thus
$\displaystyle \displaystyle \vec{v} = \left ( \frac{3i - 4j}{|3i - 4j|} \right ) \cdot 10~\text{m/s}$

-Dan

3. Thank you.