Answer
Result Moves in a straight line with constant speed.
Work Step by Step
Step 1 In this exercise, we consider the trajectory of a particle with zero acceleration in 2-space or 3-space.
Step 2 Acceleration is like a push in a particular direction. If it is in the same direction as you are traveling, you will speed up; if it is in the opposite direction, you will slow down. In other directions, it will cause you to turn. If the acceleration is zero, then a moving particle will have to be traveling in a straight line with constant speed. (A particle at rest will also have zero acceleration)
Step 3 One way to show this is to make an acceleration vector that is zero: \[ \mathbf{a}(t) = 0\mathbf{i} + 0\mathbf{j} + 0\mathbf{k} \] Everything is the same in the case for 2-space, just without the \(\mathbf{k}\) term.
Step 4 Next, we integrate acceleration to get the velocity vector. \(a, b, c\) are the constants of integration. \[ \mathbf{v}(t) = \int \mathbf{a}(t) \, dt = ai + bj + ck \] From this, we can see that the movement speed is constant.
Step 5 Then we integrate velocity to get the position vector. \(x_1, y_1, z_1\) are the constants of integration. \[ \mathbf{r}(t) = \int \mathbf{v}(t) \, dt = (x_1 + at)i + (y_1 + bt)j + (z_1 + ct)k \] The components may look familiar as the parametric equations of a line: \[ \begin{align*} x &= x_1 + at \\ y &= y_1 + bt \\ z &= z_1 + ct \end{align*} \] So, a particle moving with zero acceleration is moving in a straight line with constant speed.
Step 6 Summary By starting with a zero acceleration vector, we integrated it once to get velocity and saw that the speed is constant. Then we integrated velocity to get the position vector and saw that it moved in a straight line. Result Moves in a straight line with constant speed.
