The problem presented is an initial value problem involving a differential equation modeling the velocity of a person who has bailed out of a helicopter with a parachute, taking into account the effect of air resistance.

The equation given is:

$\frac{dv}{dt} = 32 - 1.76v, \quad v(0) = 0$

where:

• $v(t)$ is the velocity in feet per second (ft/s),
• $t$ is the time in seconds,
• The constant $k = 1.76$ represents the air resistance proportionality constant,
• The term 32 represents the acceleration due to gravity in ft/s².

Part 1: Limiting Velocity

The limiting velocity is the velocity that the person will approach as time $t$ becomes large. This occurs when the velocity stops changing, i.e., when $\frac{dv}{dt} = 0$.

Solving for $v$ when $\frac{dv}{dt} = 0$:

$0 = 32 - 1.76v$ $v = \frac{32}{1.76} = 18.18 \, \text{ft/s}$

So, the limiting velocity is 18.18 ft/s.

Part 2: Time to Reach 95% of Limiting Velocity

Next, we need to determine how long it will take for the person to reach 95% of the limiting velocity. Let’s solve the differential equation to get the velocity as a function of time.

Solving the Differential Equation

The equation is separable, so we can solve it by separating variables:

$\frac{dv}{32 - 1.76v} = dt$

Integrating both sides:

$\int \frac{1}{32 - 1.76v} \, dv = \int 1 \, dt$

The left-hand side is a standard logarithmic integral:

$-\frac{1}{1.76} \ln |32 - 1.76v| = t + C$

Exponentiating both sides and solving for $v(t)$, we obtain the velocity as a function of time. To get the particular solution, we apply the initial condition $v(0) = 0$. From this, we can find the time it takes to reach 95% of the limiting velocity, which is $0.95 \times 18.18 = 17.27$ ft/s.

Let me calculate the exact time it takes to reach 95% of the limiting velocity.The real solution for the time it takes to reach 95% of the limiting velocity is approximately:

$t = 1.70 \, \text{seconds}$

This means it will take about 1.7 seconds to reach 95% of the limiting velocity (which is approximately 17.27 ft/s).

Recap and Further Questions:

1. Limiting Velocity: The limiting velocity is 18.18 ft/s.
2. Time to 95% Limiting Velocity: You will reach 95% of the limiting velocity (17.27 ft/s) in approximately 1.7 seconds.

Would you like more details on any step? Here are some further questions you might explore:

1. What happens if the air resistance constant $k$ increases?
2. How does the initial condition $v(0) = 0$ affect the solution?
3. Could you find the time to reach other percentages of the limiting velocity, like 99%?
4. What would be the behavior of the system if there was no air resistance?
5. How would a different gravitational constant, such as on another planet, impact the limiting velocity?

Tip:

When dealing with separable differential equations, always remember to isolate the variables and integrate both sides step by step carefully.