3 In this question the origin is a point on the ground. The directions of the unit vectors \(\left( \begin{array} { l } 1
0
0 \end{array} \right) , \left( \begin{array} { l } 0
1
0 \end{array} \right)\) and \(\left( \begin{array} { l } 0
0
1 \end{array} \right)\) are
east, north and vertically upwards.
\includegraphics[max width=\textwidth, alt={}, center]{cb72a1c4-f769-4348-ad7f-66c3c96e1732-3_401_686_368_721}
Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.
- During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
- During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
- Alesha's mass, including her equipment, is 100 kg .
- Initially, her position vector is \(\left( \begin{array} { r } - 75
90
750 \end{array} \right) \mathrm { m }\) and her velocity is \(\left( \begin{array} { r } - 5
0
- 10 \end{array} \right) \mathrm { ms } ^ { - 1 }\).
- Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.
At a certain time during the dive, forces of \(\left( \begin{array} { r } 0
0
- 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0
0
880 \end{array} \right) \mathrm { N }\) and \(\left( \begin{array} { r } 50
- 20
0 \end{array} \right) \mathrm { N }\) are acting on Alesha.
Suggest how these forces could arise.Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is \(1.14 \mathrm {~ms} ^ { - 2 }\).
One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.Find expressions for her velocity and position vector at time \(t\) seconds after the start of the dive according to this model. Verify that when \(t = 30\) she is at the origin.Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.