Answer:
The velocity is [tex]4.6 m/s^2[/tex]
Explanation:
Given:
Force = 500N
Distance s= 0
To find :
Its velocity at s = 0.5 m
Solution:
[tex]\sum F_{x}=m a[/tex]
[tex]F\left(\frac{4}{5}\right)-F_{S}=13 a[/tex]
[tex]500\left(\frac{4}{5}\right)-\left(k_{s}\right)=13 a[/tex]
[tex]400-(500 s)=13 a[/tex]
[tex]a = \frac{400 -(500s)}{13}[/tex]
[tex]a = (30.77 -38.46s) m/s^2[/tex]
Using the relation,
[tex]a=\frac{d v}{d t}=\frac{d v}{d s} \times \frac{d s}{d t}[/tex]
[tex]a=v \frac{d v}{d s}[/tex]
[tex]v d v=a d s[/tex]
Now integrating on both sides
[tex]\int_{0}^{v} v d v=\int_{0}^{0.5} a d s[/tex]
[tex]\int_{0}^{v} v d v=\int_{0}^{0.5}(30.77-38.46 s) d s[/tex]
[tex]\left[\frac{v^{2}}{2}\right]_{0}^{2}=\left[\left(30.77 s-19.23 s^{2}\right)\right]_{0}^{0.5}[/tex]
[tex]\left[\frac{v^{2}}{2}\right]=\left[\left(30.77(0.5)-19.23(0.5)^{2}\right)\right][/tex]
[tex]\left[\frac{v^{2}}{2}\right]=[15.385-4.807][/tex]
[tex]\left[\frac{v^{2}}{2}\right]=10.578[/tex]
[tex]v^{2}=10.578 \times 2[/tex]
[tex]v^{2}=21.15[/tex]
[tex]v = \sqrt{21.15}[/tex]
[tex]v = 4.6 m/s^2[/tex]
