Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problema 6.R.5

Capitolo 6, Problema 6.R.5

Position, displacement, and distance A projectile is launched vertically from the ground at t=0, and its velocity in flight (in m/s) is given by v(t)=20−10t. Find the position, displacement, and distance traveled after t seconds, for 0≤t≤4.

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Identify the given velocity function: \(v(t) = 20 - 10t\), which represents the velocity of the projectile at time \(t\) seconds.

Find the position function \(s(t)\) by integrating the velocity function with respect to time: \(s(t) = \int v(t) \, dt = \int (20 - 10t) \, dt\). Remember to include the constant of integration and use the initial condition \(s(0) = 0\) since the projectile starts from the ground.

Calculate the displacement after \(t\) seconds by evaluating the position function at \(t=4\): \(\text{Displacement} = s(4) - s(0)\). Since \(s(0) = 0\), this simplifies to \(s(4)\).

Determine the distance traveled by considering the total length of the path. Since velocity changes sign, find the time when \(v(t) = 0\) to identify when the projectile changes direction: solve \(20 - 10t = 0\) for \(t\). Then, calculate the position at this time and use it to find the total distance traveled by summing the absolute values of position changes over each interval.

Summarize the results: position function \(s(t)\) for \(0 \leq t \leq 4\), displacement as \(s(4)\), and distance traveled as the sum of absolute position changes considering the change in direction.

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Velocity and Its Relation to Position

Velocity is the rate of change of position with respect to time. Given a velocity function v(t), the position function s(t) can be found by integrating v(t) over time, considering initial conditions. This relationship allows us to determine the object's location at any time t.

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Intro To Related Rates

Displacement vs. Distance Traveled

Displacement is the net change in position from the starting point, which can be positive, negative, or zero. Distance traveled is the total length of the path covered, always non-negative, and accounts for all movement regardless of direction. Understanding the difference is crucial when velocity changes sign.

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Using The Acceleration Function Example 1

Integration and Absolute Value in Calculating Distance

To find distance traveled from velocity, integrate the absolute value of velocity over the time interval. This accounts for changes in direction by summing all movement magnitudes. Identifying when velocity changes sign helps split the integral correctly for accurate distance calculation.

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Initial Value Problems Example 1

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