Question

Translate the following real-world scenario into signals. Write mathematically.

A parachute jumper with mass 80 kg rides in a plane at 4000 m above sea level (ASL). At an arbitrary time zero, the jumper jumps out of the plane. At a height of 750 m, she deploys her parachute, after which she keeps a constant vertical velocity of 8m/s. Ignoring air resistance for the portion of her drop before parachute deployment, write an analytical expression of her height as a function of time with correct units.

This is for my signals and systems class. Thank you!

Answer 1

Answer 2

To translate the given real-world scenario into signals, we can express the height of the parachute jumper as a function of time. Let's break it down step by step.

Step 1: Identify the key information:

• Mass of the parachute jumper: m = 80 kg

• Initial height above sea level: h0 = 4000 m

• Height at parachute deployment: h_parachute = 750 m

• Vertical velocity after parachute deployment: v = 8 m/s

Step 2: Determine the time intervals: We can divide the scenario into two time intervals: before parachute deployment (t < t_parachute) and after parachute deployment (t >= t_parachute).

Step 3: Analyze the height function for each interval: Interval 1: Before Parachute Deployment (t < t_parachute) Since we are ignoring air resistance for this portion, the jumper's height decreases due to the acceleration due to gravity (g = 9.8 m/s²). We can use the equation of motion for free fall to describe the height:

h1(t) = h0 - (1/2)gt²

Interval 2: After Parachute Deployment (t >= t_parachute) Once the parachute is deployed, the jumper maintains a constant vertical velocity of 8 m/s. The height change in this interval can be described by:

h2(t) = h_parachute + v(t - t_parachute)

Step 4: Combine the height functions for both intervals: To get the complete height function, we can piece together the expressions for each interval:

h(t) = h1(t), for t < t_parachute h(t) = h2(t), for t >= t_parachute

Therefore, the analytical expression for the height as a function of time (t) with correct units is:

h(t) = h0 - (1/2)gt², for t < t_parachute h_parachute + v(t - t_parachute), for t >= t_parachute

Make sure to substitute the appropriate values for t_parachute, g, h0, h_parachute, and v in the expressions.

Please note that this expression assumes ideal conditions and neglects factors such as air resistance, variations in gravitational acceleration, and other external forces that may affect the actual behavior of the parachute jumper