2. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A particle is moving along a straight line. At time t seconds, tgt 0 the velocity of the particle is vms^-1 where v=2t-7sqrt (t)+6 (a) Find the acceleration of the particle when t=4 When t=1 the particle is at the point X. When t=2 the particle is at the point Y. Given that the particle does not come to instantaneous rest in the interval 1lt tlt 2 (b) show that XY=(1)/(3)(41-28sqrt (2)) metres.

# Explanation:## Step 1:To find the acceleration, we need to differentiate the velocity function, \(v(t)\), with respect to time. This gives us the acceleration function, \(a(t)\).### \(a(t) = \frac{dv}{dt} = \frac{d}{dt}(2t - 7\sqrt{t} + 6)\)## Step 2:Differentiate each term of the velocity function:### \(\frac{d}{dt}(2t) = 2\)### \(\frac{d}{dt}(-7\sqrt{t}) = -7 \cdot \frac{1}{2} t^{-1/2} = -\frac{7}{2} t^{-1/2}\)### \(\frac{d}{dt}(6) = 0\)## Step 3:Combine the differentiated terms to get the acceleration function:### \(a(t) = 2 - \frac{7}{2} t^{-1/2}\)## Step 4:To find the acceleration when , substitute into the acceleration function:### \(a(4) = 2 - \frac{7}{2} \cdot 4^{-1/2} = 2 - \frac{7}{2} \cdot \frac{1}{2} = 2 - \frac{7}{4} = \frac{8}{4} - \frac{7}{4} = \frac{1}{4}\)## Step 5:To find the displacement , we need to integrate the velocity function from to :### \(s(t) = \int (2t - 7\sqrt{t} + 6) \, dt\)## Step 6:Integrate each term of the velocity function:### ### ### ## Step 7:Combine the integrated terms to get the position function:### \(s(t) = t^2 - \frac{14}{3} t^{3/2} + 6t + C\)## Step 8:To find the constant , use the initial condition \(s(1) = X\):### ### ### ### ### ## Step 9:To find the position at , use the position function:### ### ### ## Step 10:The displacement is :### \(XY = \left( 16 - \frac{28\sqrt{2}}{3} + C \right) - \left( \frac{7}{3} + C \right)\)### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### \( XY = 16 - \frac{28\sqrt{2} +