(a) A bucket full of water with a mass of 8kg is pulled out of a well with a light inextensible rope. Find its acceleration when tha tension in the rope is 150N. [Take \(g = 10ms^{-2}\)].
(b) A mass of 12kg is acted upon by a force F, changing its speed from 15 m/s to 25 m/s after covering a distance of 50m. Find the :
(i) value of F ; (ii) distance covered when its speed is 35 m/s.
Explanation
(a) 
Let T be the tension in the rope and W the weight of the bucket of water.
T - W = net force = ma
\(T - mg = ma\)
\(150 - 80 = 8a\)
\(a = \frac{70}{8} = 8.75 m/s^{2}\).
(b) m = 12kg ; u = 15 m/s.
v = 25 m/s ; s = 50m ; a = ?
\(v^{2} = u^{2} + 2as\)
\(625 = 225 + 100a \implies 100a = 400\)
\(a = 4m/s^{2}\)
\(t = ?\)
\(s = ut + \frac{1}{2} at^{2}\)
\(50 = 15t + \frac{1}{2}(4t^{2})\)
\(50 = 15t + 2t^{2} \implies 2t^{2} + 15t - 50 = 0\)
\(2t^{2} + 20t - 5t - 50 = 0\)
\(2t(t + 10) - 5(t + 10) = 0 \implies (t + 10)(2t - 5) = 0\)
\(t = \frac{5}{2}s \text{or t = -10s}\)
Since time cannot be negative, t = 2.5 seconds.
\(F = m\frac{(v - u)}{t}\)
= \(12(\frac{25 - 15}{2.5})\)
= \(48N\)
\(2as = v^{2} - u^{2}\)
\(2as = 35^{2} - 15^{2}\)
\(2(4)(s) = 1000\)
\(s = 125m\)