What This Practice Set Covers
Vector-valued functions describe motion by giving the horizontal and vertical coordinates of a particle as functions of time. In AP Calculus BC, you need to move comfortably between the position function, its derivatives, and the physical meaning of those derivatives.
If the position of a particle is represented by
then differentiating component-by-component gives velocity and acceleration. The magnitude of the velocity vector gives the particle's speed.
The problems below focus on the calculations and interpretations that appear most often on AP-style questions.
Essential Formulas
Remember that velocity is a vector, while speed is a scalar. A particle can have a velocity vector such as while its speed is the positive number .
Practice Problems
A particle moves in the plane with position vector . Find the velocity vector at .
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The position of a particle is . Find its acceleration vector at .
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A particle has velocity vector . Find the speed of the particle at .
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A particle has velocity and initial position . Find the position vector .
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A particle has position . Find its speed at .
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A particle moves according to . At what time in the interval is the particle momentarily at rest?
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A particle has position . Determine whether the particle is moving to the right or left at .
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A particle has velocity and acceleration at . Is the particle speeding up or slowing down?
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A particle's position is . Find the position, velocity, and acceleration vectors at .
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A particle moves according to . Find the particle's speed at , determine whether it is speeding up or slowing down at that time, and find its acceleration vector.
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How to Approach AP Vector Problems
- Write the position, velocity, and acceleration separately. Keeping these three functions visible prevents many sign and differentiation errors.
- Differentiate component-by-component. Treat the x- and y-components as ordinary functions of time.
- Use magnitude only when the question asks for speed. Do not report a vector as the speed.
- Substitute the time value after differentiating. This usually makes the algebra cleaner and reduces mistakes.
- Interpret the signs. A positive x-component of velocity means the particle is moving right; a negative x-component means it is moving left.
- For speeding up or slowing down, compare velocity and acceleration. If , speed is increasing. If , speed is decreasing.
Common Mistakes
Velocity has both magnitude and direction and is represented by a vector. Speed is the magnitude of velocity. If , then the speed is , not the vector itself.
For , you must differentiate both and . The result is another vector-valued function.
Keep the velocity components separate until after you have substituted the requested time. Then calculate the magnitude. This avoids accidentally treating a vector like an ordinary scalar function.
Acceleration by itself does not tell you whether a particle is speeding up. Compare velocity and acceleration using their dot product, . A positive result means speeding up; a negative result means slowing down.
Key Takeaways
- The position vector describes where the particle is.
- The velocity vector describes how position is changing.
- The acceleration vector describes how velocity is changing.
- Speed is the magnitude of the velocity vector.
- To determine whether a particle is speeding up or slowing down, compare velocity and acceleration.
- On AP Calculus BC, vector-valued functions are usually less about difficult algebra and more about correctly interpreting derivatives and magnitudes.
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