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) = 9−t² on [0, 4]; s(0)=−2
10. Physics Applications of Integrals
Kinematics
2:40 minutesProblem 6.1.16a
Textbook Question
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
a. Determine when the motion is in the positive direction and when it is in the negative direction.
v(t) = 50e^−2t on [0, 4]
Verified step by step guidance1
Identify the velocity function given: $v(t) = 50e^{-2t}$, where $t$ is in the interval $[0, 4]$ seconds.
Recall that the direction of motion depends on the sign of the velocity: if $v(t) > 0$, the object moves in the positive direction; if $v(t) < 0$, it moves in the negative direction.
Analyze the expression $50e^{-2t}$. Since $50$ is positive and the exponential function $e^{-2t}$ is always positive for all real $t$, the velocity $v(t)$ is always positive on the interval $[0, 4]$.
Conclude that the object moves in the positive direction for all $t$ in $[0, 4]$ because $v(t) > 0$ throughout this interval.
Note that since $v(t)$ never becomes negative or zero (except possibly at infinity), the object does not move in the negative direction during the given time interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Direction of Motion
Velocity indicates both the speed and direction of an object's motion. A positive velocity means the object moves in the positive direction along the line, while a negative velocity means motion in the opposite direction. Understanding the sign of velocity helps determine when the object changes direction.
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Exponential Decay Function
The velocity function v(t) = 50e^(-2t) is an exponential decay, meaning the velocity decreases over time but remains positive since e^(-2t) > 0 for all t. This implies the object slows down but continues moving in the positive direction on the interval [0,4].
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Time Interval Analysis
Analyzing the velocity over a specific time interval [0,4] involves evaluating the function at various points to understand motion behavior. Since velocity remains positive throughout this interval, the object moves positively without reversing direction during this time.
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