# An object's two dimensional velocity is given by v(t) = ( 3t^2 - 5t , t ). What is the object's rate and direction of acceleration at t=2 ?

Jul 3, 2016

$\vec{a} \left(t\right) = \left(7 \hat{i} + \hat{j}\right) \frac{m}{s} ^ 2$

#### Explanation:

I'm guessing they're using normal convention replaced by brackets.

So, in vectorial form, $\vec{v \left(t\right)} = \left(3 {t}^{2} - 5 t\right) \hat{i} + t \hat{j}$

Acceleration is the time derivative of velocity, so you go that.
$\vec{a} = \frac{d}{\mathrm{dt}} \left(\vec{v} \left(t\right)\right) = \frac{d}{\mathrm{dt}} \left(3 {t}^{2} - 5 t\right) \hat{i} + \frac{d}{\mathrm{dt}} \left(t\right) \setminus \hat{j}$

I'm sure you know differentiation, hence we'll get acceleration
$\vec{a} \left(t\right) = \left(6 t - 5\right) \hat{i} + 1 \hat{j}$

We need to find acceleration at time $t = 2$, substitute,
$\vec{a} {\left(t\right)}_{t = 2} = \left(6 \cdot 2 - 5\right) \hat{i} + \hat{j}$

In the end, you'll get why the answer is as given in the box.