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:<br /><br />## Step 1:<br />To find the acceleration, we need to differentiate the velocity function, \(v(t)\), with respect to time. This gives us the acceleration function, \(a(t)\).<br />### \(a(t) = \frac{dv}{dt} = \frac{d}{dt}(2t - 7\sqrt{t} + 6)\)<br /><br />## Step 2:<br />Differentiate each term of the velocity function:<br />### \(\frac{d}{dt}(2t) = 2\)<br />### \(\frac{d}{dt}(-7\sqrt{t}) = -7 \cdot \frac{1}{2} t^{-1/2} = -\frac{7}{2} t^{-1/2}\)<br />### \(\frac{d}{dt}(6) = 0\)<br /><br />## Step 3:<br />Combine the differentiated terms to get the acceleration function:<br />### \(a(t) = 2 - \frac{7}{2} t^{-1/2}\)<br /><br />## Step 4:<br />To find the acceleration when \(t = 4\), substitute \(t = 4\) into the acceleration function:<br />### \(a(4) = 2 - \frac{7}{2} \cdot 4^{-1/2} = 2 - \frac{7}{2} \cdot \frac{1}{2} = 2 - \frac{7}{4} = \frac{8}{4} - \frac{7}{4} = \frac{1}{4}\)<br /><br />## Step 5:<br />To find the displacement \(XY\), we need to integrate the velocity function from \(t = 1\) to \(t = 2\):<br />### \(s(t) = \int (2t - 7\sqrt{t} + 6) \, dt\)<br /><br />## Step 6:<br />Integrate each term of the velocity function:<br />### \(\int 2t \, dt = t^2\)<br />### \(\int -7\sqrt{t} \, dt = -7 \cdot \frac{2}{3} t^{3/2} = -\frac{14}{3} t^{3/2}\)<br />### \(\int 6 \, dt = 6t\)<br /><br />## Step 7:<br />Combine the integrated terms to get the position function:<br />### \(s(t) = t^2 - \frac{14}{3} t^{3/2} + 6t + C\)<br /><br />## Step 8:<br />To find the constant \(C\), use the initial condition \(s(1) = X\):<br />### \(X = 1^2 - \frac{14}{3} \cdot 1^{3/2} + 6 \cdot 1 + C\)<br />### \(X = 1 - \frac{14}{3} + 6 + C\)<br />### \(X = 7 - \frac{14}{3} + C\)<br />### \(X = \frac{21}{3} - \frac{14}{3} + C\)<br />### \(X = \frac{7}{3} + C\)<br /><br />## Step 9:<br />To find the position at \(t = 2\), use the position function:<br />### \(Y = 2^2 - \frac{14}{3} \cdot 2^{3/2} + 6 \cdot 2 + C\)<br />### \(Y = 4 - \frac{14}{3} \cdot 2\sqrt{2} + 12 + C\)<br />### \(Y = 16 - \frac{28\sqrt{2}}{3} + C\)<br /><br />## Step 10:<br />The displacement \(XY\) is \(Y - X\):<br />### \(XY = \left( 16 - \frac{28\sqrt{2}}{3} + C \right) - \left( \frac{7}{3} + C \right)\)<br />### \(XY = 16 - \frac{28\sqrt{2}}{3} - \frac{7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \(XY = 16 - \frac{28\sqrt{2} + 7}{3}\)<br />### \( XY = 16 - \frac{28\sqrt{2} +