Answer:
The speed of the box after 2 s is 3 m/s.
Explanation:
The net force on the box in vertical direction:
![{F_{net}} = {F_g} - f - {F_p}\sin {45^{\rm{o}}}F](https://tex.z-dn.net/?f=%7BF_%7Bnet%7D%7D%20%3D%20%7BF_g%7D%20-%20f%20-%20%7BF_p%7D%5Csin%20%7B45%5E%7B%5Crm%7Bo%7D%7D%7DF)
Here,
is the gravitational force, f is the force of friction and
is the pushing force.
According to Newton’s second law, the net force is:
![{F_{net}} = ma](https://tex.z-dn.net/?f=%7BF_%7Bnet%7D%7D%20%3D%20ma)
Here, a is the acceleration in vertical direction
![\begin{array}{c}\\ \Rightarrow ma = {F_g} - f - {F_p}\sin {45^{\rm{o}}}\\\\a = \frac{{{F_g} - f - {F_p}\sin {{45}^{\rm{o}}}}}{m}\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%5C%5C%20%5CRightarrow%20ma%20%3D%20%7BF_g%7D%20-%20f%20-%20%7BF_p%7D%5Csin%20%7B45%5E%7B%5Crm%7Bo%7D%7D%7D%5C%5C%5C%5Ca%20%3D%20%5Cfrac%7B%7B%7BF_g%7D%20-%20f%20-%20%7BF_p%7D%5Csin%20%7B%7B45%7D%5E%7B%5Crm%7Bo%7D%7D%7D%7D%7D%7Bm%7D%5C%5C%5Cend%7Barray%7D)
Substitute 30N for Fg, 13N for f and 23N for Fp
![\begin{array}{c}\\a = \frac{{\left( {30{\rm{ N}}} \right) - \left( {13{\rm{ N}}} \right) - \left( {23{\rm{ N}}} \right)\sin {{45}^{\rm{o}}}}}{{3.1{\rm{ kg}}}}\\\\ = \frac{{0.7365{\rm{ N}}}}{{3.1{\rm{ kg}}}}\\\\ = 0.24\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%5C%5Ca%20%3D%20%5Cfrac%7B%7B%5Cleft%28%20%7B30%7B%5Crm%7B%20N%7D%7D%7D%20%5Cright%29%20-%20%5Cleft%28%20%7B13%7B%5Crm%7B%20N%7D%7D%7D%20%5Cright%29%20-%20%5Cleft%28%20%7B23%7B%5Crm%7B%20N%7D%7D%7D%20%5Cright%29%5Csin%20%7B%7B45%7D%5E%7B%5Crm%7Bo%7D%7D%7D%7D%7D%7B%7B3.1%7B%5Crm%7B%20kg%7D%7D%7D%7D%5C%5C%5C%5C%20%3D%20%5Cfrac%7B%7B0.7365%7B%5Crm%7B%20N%7D%7D%7D%7D%7B%7B3.1%7B%5Crm%7B%20kg%7D%7D%7D%7D%5C%5C%5C%5C%20%3D%200.24%5C%3B%7B%5Crm%7Bm%2F%7D%7D%7B%7B%5Crm%7Bs%7D%7D%5E%7B%5Crm%7B2%7D%7D%7D%5C%5C%5Cend%7Barray%7D)
Newton’s equation of motion:
![v - u = at](https://tex.z-dn.net/?f=v%20-%20u%20%3D%20at)
Substitute 2.5m/s for u , 0.24m/s2 for a and 2s for t .
![\begin{array}{c}\\v - 2.5{\rm{ m/s}} = \left( {0.24{\rm{ m/}}{{\rm{s}}^2}} \right)\left( {2{\rm{ s}}} \right)\\\\v = 0.48{\rm{ m/s}} + 2.5{\rm{ m/s}}\\\\ = 2.98{\rm{ m/s}}\\\\ \approx {\bf{3 m}}{\rm{/}}{\bf{s}}\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%5C%5Cv%20-%202.5%7B%5Crm%7B%20m%2Fs%7D%7D%20%3D%20%5Cleft%28%20%7B0.24%7B%5Crm%7B%20m%2F%7D%7D%7B%7B%5Crm%7Bs%7D%7D%5E2%7D%7D%20%5Cright%29%5Cleft%28%20%7B2%7B%5Crm%7B%20s%7D%7D%7D%20%5Cright%29%5C%5C%5C%5Cv%20%3D%200.48%7B%5Crm%7B%20m%2Fs%7D%7D%20%2B%202.5%7B%5Crm%7B%20m%2Fs%7D%7D%5C%5C%5C%5C%20%3D%202.98%7B%5Crm%7B%20m%2Fs%7D%7D%5C%5C%5C%5C%20%5Capprox%20%7B%5Cbf%7B3%20m%7D%7D%7B%5Crm%7B%2F%7D%7D%7B%5Cbf%7Bs%7D%7D%5C%5C%5Cend%7Barray%7D)
The speed of the box after 2 s is 3 m/s.