Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.


c. If the probe was released from an altitude of 3 km, when does it enter the ocean?

Verified step by step guidance

1. First, understand the problem: the probe falls freely with velocity \(v(t) = 9.8t\) m/s for the first 10 seconds, then slows instantly to a constant velocity of 10 m/s until it reaches the ocean, starting from an altitude of 3000 meters.
2. Calculate the distance fallen during the first 10 seconds by integrating the velocity function \(v(t) = 9.8t\) over the interval from \(t=0\) to \(t=10\). This gives the displacement \(s_1\) during free fall:
3. \[ s_1 = \int_0^{10} 9.8t \, dt \]
4. Evaluate the integral to find \(s_1\), which represents how far the probe has fallen in the first 10 seconds.
5. Determine the remaining distance to the ocean after 10 seconds by subtracting \(s_1\) from the initial altitude (3000 m). Then, find the time \(t_2\) it takes to cover this remaining distance at the constant speed of 10 m/s:
6. \[ t_2 = \frac{3000 - s_1}{10} \]
7. Finally, add the initial 10 seconds to \(t_2\) to find the total time \(T\) when the probe enters the ocean:
8. \[ T = 10 + t_2 \]

Key Concepts

Piecewise Functions in Motion

The velocity of the probe changes at a specific time, requiring the use of piecewise functions to model its motion. Before 10 seconds, velocity increases linearly with time; after 10 seconds, velocity is constant. Understanding how to handle different function definitions over intervals is essential to analyze total displacement.

Integration to Find Displacement

Displacement is found by integrating the velocity function over time. For variable velocity, the integral sums the area under the velocity-time curve, representing distance traveled. Calculating displacement before and after the chute deploys involves integrating each velocity segment separately.

Solving for Time Using Total Displacement

To find when the probe hits the ocean, set the total displacement equal to the initial altitude (3 km) and solve for time. This involves summing distances from both motion phases and solving the resulting equation, often requiring algebraic manipulation or numerical methods.