4.1.3 Constant Acceleration

If the acceleration $\vec{a} = a_{x} î + a_{y} ĵ$ is constant, then the two components $a_{x}$ and $a_{y}$ are both constant. In this case, everything you learned about constant-acceleration kinematics in 《 SECTION 2.4 carries over to two-dimensional motion.

Consider a particle that moves with constant acceleration from an initial position $\vec{r}_{i} = x_{i} î + y_{i} ĵ$ , starting with initial velocity $\vec{v}_{i} = v_{i x} î + v_{i y} ĵ$ . Its position and velocity at a final point f are

\begin{array}{l} x_{f} = x_{i} + v_{i x} Δ t + \frac{1}{2} a_{x} {(Δ t)}^{2} & y_{f} = y_{i} + v_{i y} Δ t + \frac{1}{2} a_{y} {(Δ t)}^{2} \\ v_{f x} = v_{i x} + a_{x} Δ t & v_{f y} = v_{i y} + a_{y} Δ t \end{array}

(4.11)

There are many quantities to keep track of in two-dimensional kinematics, making the pictorial representation all the more important as a problem-solving tool.

NOTE For constant acceleration, the x-component of the motion and the y-component of the motion are independent of each other. However, they remain connected through the fact that $Δ t$ must be the same for both.

EXAMPLE 4.3 Plotting a spacecraft trajectory

In the distant future, a small spacecraft is drifting “north” through the galaxy at 680 m/s when it receives a command to return to the starship. The pilot rotates the spacecraft until the nose is pointed 25° north of east, then engages the ion engine. The spacecraft accelerates at $75 {m/s}^{2}$ . Plot the spacecraft’s trajectory for the first 20 s.

MODEL Model the spacecraft as a particle with constant acceleration.

VISUALIZE FIGURE 4.7 shows a pictorial representation in which the y-axis points north and the spacecraft starts at the origin. Notice that each point in the motion is labeled with two positions (x and y), two velocity components ( $v_{x}$ and $v_{y}$ ), and the time t. This will be our standard labeling scheme for trajectory problems.

The figure illustrates a pictorial representation of the trajectory of a spacecraft in the x y coordinate plane.

FIGURE 4.7 Full Alternative Text

SOLVE The acceleration vector has both x- and y-components; their values have been calculated in the pictorial representation. But it is a constant acceleration, so we can write

\begin{matrix} x_{1} & = & x_{0} + v_{0 x} (t_{1} - t_{0}) + \frac{1}{2} a_{x} {(t_{1} - t_{0})}^{2} \\ = & 34.0 t_{1}^{2} m \\ y_{1} & = & y_{0} + v_{0 y} (t_{1} - t_{0}) + \frac{1}{2} a_{y} {(t_{1} - t_{0})}^{2} \\ = & 680 t_{1} + 15.8 t_{1}^{2} m \end{matrix}

where $t_{1}$ is in s. Graphing software produces the trajectory shown in FIGURE 4.8. The trajectory is a parabola, which is characteristic of two-dimensional motion with constant acceleration.

The figure illustrates a graph of the trajectory of a spacecraft on a coordinate plane.

FIGURE 4.8 Full Alternative Text

4.1.3 Constant Acceleration

EXAMPLE 4.3 Plotting a spacecraft trajectory

FIGURE 4.7 Pictorial representation of the spacecraft.

FIGURE 4.8 The spacecraft trajectory.