At time t = 0, a parachutist jumps out of an airplane that is travelling horizontally. The velocity, v m s^-1, of the parachutist at time t seconds is given by: v = (40e^{-0.02t})i + 50(e^{-0.02t} - 1)j The unit vectors i and j are horizontal and vertical respectively. Assume that the parachutist is at the origin when t = 0. Model the parachutist as a particle. (a) Find an expression for the position vector of the parachutist at time t. (b) The parachutist opens her parachute when she has travelled 100 metres horizontally. Find the vertical displacement of the parachutist from the origin when she opens her parachute. (c) Carefully, explaining the steps that you take, deduce the value of g used in the formulation of this model.

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At time t = 0, a parachutist jumps out of an airplane that is travelling horizontally - AQA - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Question 15

At time t = 0, a parachutist jumps out of an airplane that is travelling horizontally. The velocity, v m s^-1, of the parachutist at time t seconds is given by: v =... show full transcript

Worked Solution & Example Answer:At time t = 0, a parachutist jumps out of an airplane that is travelling horizontally - AQA - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Step 1

Find an expression for the position vector of the parachutist at time t.

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Answer

To find the position vector of the parachutist, we need to integrate the velocity function. Given:

$v = (40e^{-0.02t})i + 50(e^{-0.02t} - 1)j$

We integrate each component:

For the horizontal component: $x(t) = \int 40e^{-0.02t} dt = -200e^{-0.02t} + C_x$ Since the parachutist starts at the origin, when t = 0, x(0) = 0, thus: $0 = -200 + C_x \implies C_x = 200$ Therefore, $x(t) = -200e^{-0.02t} + 200$
For the vertical component: $y(t) = \int 50(e^{-0.02t} - 1) dt = -250e^{-0.02t} + 50t + C_y$ For the initial condition y(0) = 0, we find the constant: $0 = -250 + C_y \implies C_y = 250$ Thus, $y(t) = -250e^{-0.02t} + 50t + 250$

Combining both components, the position vector is given by: $r(t) = (200(1 - e^{-0.02t}))i + (50t - 250e^{-0.02t} + 250)j$

Step 2

Find the vertical displacement of the parachutist from the origin when she opens her parachute.

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Answer

The parachutist opens her parachute after travelling 100 meters horizontally. To find the time when this occurs, we set the horizontal equation equal to 100:

$200(1 - e^{-0.02t}) = 100$

Solving for t: $1 - e^{-0.02t} = 0.5 \implies e^{-0.02t} = 0.5\implies -0.02t = \ln(0.5) \implies t = \frac{-\ln(0.5)}{0.02}\approx 34.657$ seconds.

Now, substituting this time back into the vertical displacement equation:

$y(t) = 50t - 250e^{-0.02t} + 250$

Calculating y(34.657):

Calculate the value:
- $e^{-0.02 * 34.657} \approx 0.5$

Thus, $y(34.657) = 50(34.657) - 250(0.5) + 250\approx 1732.85 - 125 + 250 \approx 1857.85$

The vertical displacement from the origin is: $y_{displacement} = 1857.85 meters.$

Step 3

Carefully, explaining the steps that you take, deduce the value of g used in the formulation of this model.

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Answer

To deduce the value of g, we need to analyze the vertical component of the motion.

The vertical velocity is given by: $v_y = 50(e^{-0.02t} - 1)$
To find the acceleration (which affects g), we differentiate the vertical velocity with respect to time: $a_y = \frac{dv_y}{dt} = 50(-0.02e^{-0.02t}) = -10e^{-0.02t}$
As t approaches large values, e^{-0.02t} approaches 0, making the vertical acceleration approach -10 m/s², indicating the effect of gravitational acceleration.

Thus, we deduce: $g \approx 10 \text{ m/s}^2$ .

At time t = 0, a parachutist jumps out of an airplane that is travelling horizontally - AQA - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Worked Solution & Example Answer:At time t = 0, a parachutist jumps out of an airplane that is travelling horizontally - AQA - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Find an expression for the position vector of the parachutist at time t.

Find the vertical displacement of the parachutist from the origin when she opens her parachute.

Carefully, explaining the steps that you take, deduce the value of g used in the formulation of this model.

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