The acceleration vector of a particle with position r(t) = (x(t), y(t)) is which of the following?

• A. (x''(t), y''(t))
• B. (dx/dt, dy/dt)
• C. (d^3x/dt^3, d^3y/dt^3)
• D. (x(t), y(t))

Answer: (A)

Acceleration is the rate of change of velocity, so for a particle with position r(t) = (x(t), y(t)), its velocity is v(t) = r'(t) = (x'(t), y'(t)). The acceleration is the derivative of velocity, a(t) = v'(t) = r''(t) = (x''(t), y''(t)). Thus the acceleration vector has components given by the second-time derivatives of the coordinates. The first derivatives (x'(t), y'(t)) are velocity, the original position (x(t), y(t)) is position, and the third derivatives (x'''(t), y'''(t)) relate to jerk.