Textbook Question
Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.
c. How far has the cyclist traveled when her velocity is 250 m/min?
10. Physics Applications of Integrals
Kinematics
2:36 minutesProblem 6.1.18a
Textbook Question
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = −t³+3t²−2t on [0, 3]; s(0)=4
Verified step by step guidance1
Identify the given velocity function: $v(t) = -t^{3} + 3t^{2} - 2t$ and the initial position $s(0) = 4$.
Recall that the position function $s(t)$ is the antiderivative (indefinite integral) of the velocity function $v(t)$, plus a constant of integration $C$:
$s(t) = \int v(t) \, dt + C$.
Integrate the velocity function term-by-term:
$\int (-t^{3} + 3t^{2} - 2t) \, dt = \int -t^{3} \, dt + \int 3t^{2} \, dt - \int 2t \, dt$.
Calculate each integral separately:
$\int -t^{3} \, dt = -\frac{t^{4}}{4}$,
$\int 3t^{2} \, dt = t^{3}$,
$\int -2t \, dt = -t^{2}$.
Combine the results and add the constant $C$:
$s(t) = -\frac{t^{4}}{4} + t^{3} - t^{2} + C$. Use the initial condition $s(0) = 4$ to solve for $C$ by substituting $t=0$ and $s(0)=4$.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Position Relationship
Velocity is the rate of change of position with respect to time. The position function can be found by integrating the velocity function over time, which accumulates the changes in position. Understanding this relationship is essential to move from velocity to position.
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Antiderivative (Indefinite Integral)
The antiderivative of a function is another function whose derivative is the original function. To find the position from velocity, you compute the antiderivative of the velocity function, adding a constant of integration that can be determined using initial conditions.
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Initial Conditions and Constants of Integration
When integrating to find position, an unknown constant appears. The initial position value, such as s(0) = 4, allows you to solve for this constant, ensuring the position function accurately reflects the object's starting point.
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