4.2 Acceleration Vector

Suppose the acceleration function has the form $\overset{\rightarrow}{a} \left(t\right) = a \hat{i} + b \hat{j} + c \hat{k} \text{m}/ \text{s}^{2} ,$ where a, b, and c are constants. What can be said about the functional form of the velocity function?

If an object has a constant x-component of the velocity and suddenly experiences an acceleration in the y direction, does the x-component of its velocity change?

The position of a particle for t > 0 is given by $\overset{\rightarrow}{r} \left(t\right) = \left(3.0 t^{2} \hat{i} - 7.0 t^{3} \hat{j} - 5.0 t^{−2} \hat{k}\right) \text{m} .$ (a) What is the velocity as a function of time? (b) What is the acceleration as a function of time? (c) What is the particle’s velocity at t = 2.0 s? (d) What is its speed at t = 1.0 s and t = 3.0 s? (e) What is the average velocity between t = 1.0 s and t = 2.0 s?

A particle has a position function $\overset{\rightarrow}{r} \left(t\right) = \text{cos} \left(1.0 t\right) \hat{i} + \text{sin} \left(1.0 t\right) \hat{j} + t \hat{k} ,$ where the arguments of the cosine and sine functions are in radians. (a) What is the velocity vector? (b) What is the acceleration vector?