PHSX 220 Physics I (w/ Calculus) · Unit 1: Kinematics & Newton’s laws · on Exam 1
Ch 4.1–4.2 Motion in two dimensions
Study notes for Motion in two dimensions in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Position, velocity and acceleration are vectors: $\vec r = x\hat i + y\hat j$, $\vec v = d\vec r/dt$, $\vec a = d\vec v/dt$. The $x$ and $y$ motions are independent: solve each with the 1D equations, linked only by time.
- The velocity vector is tangent to the path. The acceleration vector points toward the inside of a curve; it has a component along the velocity (changing speed) and one perpendicular (changing direction).
- Average velocity is displacement over time (a vector); average speed is distance over time (a scalar). On a round trip the average velocity is zero.
Key formulas
Vector kinematics
$$\vec v_{\text{avg}} = \frac{\Delta\vec r}{\Delta t},\quad \vec a_{\text{avg}} = \frac{\Delta\vec v}{\Delta t},\quad \vec v = \frac{d\vec r}{dt},\quad \vec a = \frac{d\vec v}{dt}$$
Constant acceleration, each axis
$$x = x_0 + v_{0x}t + \tfrac12 a_xt^2,\qquad y = y_0 + v_{0y}t + \tfrac12 a_yt^2$$
Worked example
A drone moves from $(2, 5)$ m to $(14, 0)$ m in 4.0 s. Find its average velocity.
Show the solution
$\Delta\vec r = (12, -5)$ m, so $\vec v_{\text{avg}} = (3.0, -1.25)$ m/s; magnitude $\sqrt{9 + 1.5625} = 3.25$ m/s, direction $\tan^{-1}(-1.25/3) = -22.6^\circ$ (below the $+x$ axis).
Common mistakes
- Mixing an x-equation with a y-quantity. Keep two columns.
- Assuming acceleration is along the velocity; in curved motion it is not.
Exam tipTime is the only quantity shared by the x and y columns. If you find t from one axis, carry it to the other.
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