Final answer:
The velocity of the particle is v(t) = -t, the acceleration is a constant a(t) = -1, and the speed, which is the absolute value of the velocity, equals t for positive t values.
Step-by-step explanation:
To find the velocity, acceleration, and speed of a particle with the given position function r(t) = -1/2 t², we need to differentiate the position function with respect to time to get the velocity, and then differentiate the velocity to get the acceleration.
The velocity v(t) is the first derivative of the position function r(t), so v(t) = dr(t)/dt. For the given function, this would be v(t) = d(-1/2 t²)/dt = -t.
The acceleration a(t) is the derivative of the velocity, so a(t) = dv(t)/dt. For our velocity function, the acceleration would be a(t) = d(-t)/dt = -1, which is a constant.
The speed of the particle is the magnitude of the velocity vector. Since we have a one-dimensional motion here, speed is just the absolute value of velocity, so speed = |v(t)| = | -t | = t, when t is positive.