How to represent the position?

Question

this will probably sound elementary but I had to ask.

here's an example because i can't put it better.

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say, the position is defined with $ x = 2t $ on the horizental and $ y = \frac{t^{2}}{7}$

find velocity & acceleration in magnitude and direction

now can i :

1. define a function $ S(t) $ which includes the $x$ and $y$ somehow ?

( i don't know how to do that )

1. say that the position is a vector defined by $ 2t $ i + $ \frac{t^{2}}{7}$ j

2. both are valid ?

knowing that i will take $ \frac{d}{dt}$ and carry on with my problem.

Answer 1

hint

$\vec {i} $ and $\vec {j} $ are constant.

the position is $$\vec {OM}=2t\vec {i}+\frac {t^2}{7}\vec {j} $$

by differentiation we get the velocity

$$\vec {V}(t)=\frac {d\vec {OM}}{dt}=2\vec {i}+\frac {2t}{7}\vec {j} $$ and the acceleration

$$\vec {\gamma}(t)=\frac {d^2\vec {OM}}{dt^2}=\frac {2}{7}\vec {j} $$

The magnitudes are $$\|\vec {V}(t)\|=2\sqrt {1+\frac {t^2}{49}} $$ and $$\|\vec {\gamma}(t)\|=\frac {2}{7} $$

Answer 2

You can use the vector notation that you have already derived to write the position as a function:

$$ S(t) = 2t \mathbf i + \frac{t^{2}}{7} \mathbf j .$$