Respuesta :
Answer:
(a) k = 0.09 s⁻¹
(b) The velocity= ± 16.97 mm/s
Explanation:
(a) Given that the acceleration = a = k(100 - x)
Therefore;
[tex]a = \dfrac{dv}{dt} = \dfrac{dv}{dx} \times \dfrac{dx}{dt} = \dfrac{dv}{dx} \times v = k(100 - x)[/tex]
When x = 40 mm, v = 0 mm/s hence;
[tex]\int\limits^v_0 {v } \, dv = \int\limits^x_{40} {k(100 - x)} \, dx[/tex]
[tex]\dfrac{1}{2} v^2 = k \cdot \left [100\cdot x-\frac{1}{2}\cdot x^{2} \right ]_{x}^{40}[/tex]
[tex]\dfrac{1}{2} v^2 = -\dfrac{ k\cdot \left (x^{2}-200\cdot x+6400 \right ) }{2}[/tex]
At x = 100 mm, v = 18 mm/s hence we have;
[tex]\dfrac{1}{2} 18^2 = -\dfrac{ k\cdot \left (100^{2}-200\times 100+6400 \right ) }{2} = 1800\cdot k[/tex]
[tex]\dfrac{1}{2} 18^2 =162 = 1800\cdot k[/tex]
k = 162/1800 = 9/100 = 0.09 s⁻¹
(b) When x = 120 mm, we have
[tex]\dfrac{1}{2} v^2 = -\dfrac{ 0.09\times \left (120^{2}-200\times 120+6400 \right ) }{2} = 144[/tex]
Therefore;
v² = 2 × 144 = 288
The velocity, v = √288 = ±12·√2 = ± 16.97 mm/s.
