Final answer:
To find the velocity, differentiate the position function with respect to time. The velocity is given by v(t) = i + 2t j. To find the acceleration, differentiate the velocity function with respect to time. The acceleration is given by a(t) = 2j. The speed is the magnitude of velocity, which in this case is given by speed = √(1+4t^2).
Step-by-step explanation:
The position function given is r(t) = t i + t² j + 5k. To find the velocity, we differentiate the position function with respect to time. So, v(t) = d/dt (t i + t² j + 5 k). Differentiating each component of the position function independently, we get v(t) = i + 2t j + 0 k. Therefore, the velocity function is v(t) = i + 2t j.
To find the acceleration, we differentiate the velocity function with respect to time. So, a(t) = d/dt (i + 2t j). The derivative of i with respect to time is zero and the derivative of 2t j with respect to time is 2 j. Therefore, the acceleration function is a(t) = 2j.
Speed is the magnitude of velocity. In this case, the speed is determined by evaluating the magnitude of the velocity function at any point. Since the velocity function is v(t) = i + 2t j, the speed is given by speed = |v(t)| = √(1^2 + (2t)^2) = √(1+4t^2).