1. Given data

• Initial position: origin

• Initial velocity: $\vec{u} = 5\hat{i}\,\text{m/s}$

  So, $u_x=5,\quad u_y=0$

• Constant acceleration: $\vec{a}=(3\hat{i}+2\hat{j})\,\text{m/s}^2$

  So, $a_x=3,\quad a_y=2$

We need the speed when the x-coordinate becomes $84\,$m.

2. Use x-motion to find time

For motion along $x$-axis, $x=u_xt+\frac{1}{2}a_xt^2$

Substitute values: $84=5t+\frac{1}{2}(3)t^2$ $84=5t+\frac{3}{2}t^2$

Multiply by 2: $168=10t+3t^2$ $3t^2+10t-168=0$

Solve the quadratic: $t=\frac{-10\pm\sqrt{10^2+4\cdot 3\cdot 168}}{2\cdot 3}$ $t=\frac{-10\pm\sqrt{100+2016}}{6}$ $t=\frac{-10\pm\sqrt{2116}}{6}$ $t=\frac{-10\pm 46}{6}$

Physical root: $t=\frac{36}{6}=6\,\text{s}$

3. Find velocity components at $t=6$ s

Velocity in x-direction: $v_x=u_x+a_xt=5+3(6)=23$

Velocity in y-direction: $v_y=u_y+a_yt=0+2(6)=12$

So the velocity vector is $\vec{v}=23\hat{i}+12\hat{j}$

4. Compute speed

Speed is magnitude of velocity: $v=\sqrt{v_x^2+v_y^2}$ $v=\sqrt{23^2+12^2}$ $v=\sqrt{529+144}$ $v=\sqrt{673}$

Thus, $\alpha=673$

5. Comparison with stored answer

Derived answer: $673$

Stored correct answer: $673$

They match.