3. Weather Balloons Every year, thousands of high-altitude sounding balloons, frequently called weather balloons even when...

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3. Weather Balloons Every year, thousands of high-altitude sounding balloons, frequently called weather balloons even when their purpose is not weather-related, are released from ground stations. As they are released in the sky, because they are very light they start to rise vertically, but they also move horizontally with the local wind speed. You wish to try and derive the position and acceleration of a balloon using ground measurements of the wind velocity.. You and a colleague measure the wind velocity at two stationary points in a field where the balloons are launched. In the first diagram are sensor locations and wind measurements at each location when the balloon is released (t = 0 sec). There was a gust of wind shortly after release and the data at 10 seconds is given in the second diagram. x=0m t=0 s V = 5 m/s x = 0 m t = 10 s V=8 m/s x = 200 m t=0 s V 8 m/s x = 200 m t = 10 s V = 11 m/s Figure 2: The balloon and measured wind speeds (dimensions not to scale) Use the following assumptions as part of your analysis: The balloon is very light and moves with the same velocity as the wind around it. The second sensor location is directly downwind of the first sensor. That the velocity changes linearly in space between x = 0 and x = 200 m and that there is no vertical gradient in the velocity over the times and balloon heights you are analyzing. That the velocity changes linearly in time between t = 0 sec and t = 10 sec. The second sensor being directly downwind of the first sensor, i.e. there is no y-velocity component. You are only analyzing horizontal motion, not the vertical lift and rise of the balloon. Answer the following questions: a) Write an expression for the velocity field at any point x (between the two wind velocity sensors) in the fluid at any time t. b) Write the expression for the local acceleration at a point x (between the two wind velocity sensors) c) What is the expression for the convective acceleration at a point x (between the two wind velocity sensors)? d) What is the expression for the total or material derivative (total acceleration) of the velocity as a function of x and t? e) What is the (horizontal) velocity and acceleration of the balloon at time, t = 0 sec if it is located at x = 0 m (i.e. what would an accelerometer housed inside the balloon record for the acceleration value, and what is the horizontal ground speed of the balloon)? 3. Weather Balloons Every year, thousands of high-altitude sounding balloons, frequently called weather balloons even when their purpose is not weather-related, are released from ground stations. As they are released in the sky, because they are very light they start to rise vertically, but they also move horizontally with the local wind speed. You wish to try and derive the position and acceleration of a balloon using ground measurements of the wind velocity.. You and a colleague measure the wind velocity at two stationary points in a field where the balloons are launched. In the first diagram are sensor locations and wind measurements at each location when the balloon is released (t = 0 sec). There was a gust of wind shortly after release and the data at 10 seconds is given in the second diagram. x=0m t=0 s V = 5 m/s x = 0 m t = 10 s V=8 m/s x = 200 m t=0 s V 8 m/s x = 200 m t = 10 s V = 11 m/s Figure 2: The balloon and measured wind speeds (dimensions not to scale) Use the following assumptions as part of your analysis: The balloon is very light and moves with the same velocity as the wind around it. The second sensor location is directly downwind of the first sensor. That the velocity changes linearly in space between x = 0 and x = 200 m and that there is no vertical gradient in the velocity over the times and balloon heights you are analyzing. That the velocity changes linearly in time between t = 0 sec and t = 10 sec. The second sensor being directly downwind of the first sensor, i.e. there is no y-velocity component. You are only analyzing horizontal motion, not the vertical lift and rise of the balloon. Answer the following questions: a) Write an expression for the velocity field at any point x (between the two wind velocity sensors) in the fluid at any time t. b) Write the expression for the local acceleration at a point x (between the two wind velocity sensors) c) What is the expression for the convective acceleration at a point x (between the two wind velocity sensors)? d) What is the expression for the total or material derivative (total acceleration) of the velocity as a function of x and t? e) What is the (horizontal) velocity and acceleration of the balloon at time, t = 0 sec if it is located at x = 0 m (i.e. what would an accelerometer housed inside the balloon record for the acceleration value, and what is the horizontal ground speed of the balloon)?