Sketching velocity-time graphs

1 \includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-2_675_1380_255_379} A woman walks in a straight line. The woman's velocity \(t\) seconds after passing through a fixed point \(A\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The graph of \(v\) against \(t\) consists of 4 straight line segments (see diagram). The woman is at the point \(B\) when \(t = 60\). Find

1. the woman's acceleration for \(0 < t < 30\) and for \(30 < t < 40\),
2. the distance \(A B\),
3. the total distance walked by the woman.

6 The speed of a 100 metre runner in \(\mathrm { ms } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h] \end{figure}

1. Use Model P to estimate
   (A) the distance the runner has gone at the end of 12 seconds,
   (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 }$$
2. Verify that Model Q gives the correct speed for \(t = 8\).
3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time?
5. Find the greatest acceleration of the runner according to each model.

A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(-20\) m s\(^{-1}\). The acceleration \(a\) m s\(^{-2}\) at time \(t\) s after leaving \(O\) is given by \(a = 12 - 2t\).

1. Sketch a velocity-time graph for \(0 \leq t \leq 12\), indicating the times when \(P\) is at rest. [5]
2. Find the total distance travelled by \(P\) in the interval \(0 \leq t \leq 12\). [5]

A particle \(X\) travels in a straight line. The velocity of \(X\) at time \(t\) s after leaving a fixed point \(O\) is denoted by \(v\) m s\(^{-1}\), where $$v = -0.1t^3 + 1.8t^2 - 6t + 5.6.$$ The acceleration of \(X\) is zero at \(t = p\) and \(t = q\), where \(p < q\).

1. Find the value of \(p\) and the value of \(q\). [4]

It is given that the velocity of \(X\) is zero at \(t = 14\).

1. Find the velocities of \(X\) at \(t = p\) and at \(t = q\), and hence sketch the velocity-time graph for the motion of \(X\) for \(0 \leq t \leq 15\). [3]
2. Find the total distance travelled by \(X\) between \(t = 0\) and \(t = 15\). [5]

A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O\) 100 s later. The velocity of \(P\) is \(v \text{ m s}^{-1}\) at time \(t\) s after leaving \(O\), where $$v = 0.0001t^3 - 0.015t^2 + 0.5t.$$

1. Show that \(P\) is instantaneously at rest when \(t = 0\), \(t = 50\) and \(t = 100\). [2]
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocity-time graph for \(P\)'s motion for \(0 \leq t \leq 100\). [7]
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leq t \leq 100\). [4]