91影视

• The mass of a particle is,$m=3.00kg$
• The velocity of the particle is,$\stackrel{鈫�}{v}=\left(800t\right)\hat{i}+\left(300t^2\right)\hat{j}$

The expression for velocity can be differentiated to find out acceleration. By using Newton鈥檚 second law, the time at which this net force acting on a particle can be found. By using the trigonometry formula, their directions can also be found.

Formulae:

$$\begin{gathered} \stackrel{鈫�}{F}net=m\stackrel{鈫�}{a} \\ \stackrel{鈫�}{a}=\frac{d\stackrel{鈫�}{v}}{dt} \\ 胃={\tan }^{鈥�1}\frac{a_y}{a_x} \\ 胃={\tan }^{鈥�1}\frac{v_y}{v_x} \end{gathered}$$

According to the expression of acceleration,

$\begin{array}{l}\stackrel{鈫�}{a}=\frac{d(8.00t)\hat{i}+\left(300t^2\right)\hat{j}}{dt}\\ =\left(800\right)\hat{i}+\left(6.00t\right)\hat{j}\end{array}$

According to the Newton鈥檚 second law,

localid="1657160200380" $$\begin{gathered} \stackrel{鈫�}{F}net=m\stackrel{鈫�}{a} \\ =m\left(800\right)\hat{i}+\left(6.00t\right)\hat{j} \end{gathered}$$

We have to find the time for that net force on the particle. We can take the square of magnitude of both sides and simplify it for width="6">t

localid="1657160570702" $\begin{array}{l}{F}^2=m^2\left[8.{00}^2+{(6.00t)}^2\right]\\ {35}^2=3.{00}^2\left[8.{00}^2+{(6.00t)}^2\right]\\ t=1.415s\end{array}$

We can find the acceleration as,

$\begin{array}{l}\stackrel{鈫�}{a}=\left(8.00\right)\hat{i}+\left(6.00脳1.415\right)\hat{j}\\ =\left(8.00\right)\hat{i}+\left(8.49\right)\hat{j}\end{array}$

The acceleration makes an angle with positive x axis as,

$$\begin{gathered} 胃={\tan }^{鈥�1}\frac{a_y}{a_x} \\ ={\tan }^{鈥�1}\frac{(8.49)}{(8.00)} \\ =46.7^0 \end{gathered}$$

Hence, acceleration makes an angle of $46.7^0$with the positive x axis.

The velocity vector at instant t is,

$\begin{array}{l}\stackrel{鈫�}{V}=(8.00脳1.415)\hat{i}+\left(3.001.{415}^2\right)\hat{j}\\ =\left(11.3\right)\hat{i}+\left(6.01\right)\hat{j}\end{array}$

The angle made by with positive x axis as,

$$\begin{gathered} 胃={\tan }^{鈥�1}\frac{v_y}{v_x} \\ ={\tan }^{鈥�1}\frac{6.01}{11.3} \\ =28.0^0 \end{gathered}$$

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