PHSX 220 Physics I (w/ Calculus) · Unit 1: Kinematics & Newton’s laws · on Exam 1
Ch 3.1–3.4 Motion in one dimension
Study notes for Motion in one dimension 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 StudyCat.
Big ideas
- Position $x(t)$, displacement $\Delta x = x_f - x_i$ (can be negative; distance cannot). Average velocity $\bar v = \Delta x/\Delta t$; instantaneous velocity $v = dx/dt$ (slope of the $x$–$t$ graph). Acceleration $a = dv/dt$ (slope of the $v$–$t$ graph).
- Graphs: slope of $x$–$t$ is velocity; slope of $v$–$t$ is acceleration; area under $v$–$t$ is displacement; area under $a$–$t$ is change in velocity.
- Constant acceleration gives four equations. Each one is missing one variable ($x$, $v$, $a$ or $t$); pick the equation that omits the quantity you neither know nor want.
- Speeding up means $v$ and $a$ have the same sign; slowing down means opposite signs. Negative acceleration is not the same as slowing down.
- With calculus: $v(t) = v_0 + \int_0^t a\,dt$, $x(t) = x_0 + \int_0^t v\,dt$.
Key formulas
Definitions
$$\bar v = \frac{\Delta x}{\Delta t},\quad v = \frac{dx}{dt},\quad \bar a = \frac{\Delta v}{\Delta t},\quad a = \frac{dv}{dt}$$
Constant acceleration
$$v = v_0 + at,\quad x = x_0 + v_0t + \tfrac12 at^2,\quad v^2 = v_0^2 + 2a\,\Delta x,\quad \Delta x = \tfrac12(v_0 + v)\,t$$
Worked example
A car accelerates from rest at $2.5\ \text{m/s}^2$ for 8.0 s, then brakes uniformly to a stop in 40 m. Find the braking deceleration and the total time.
Show the solution
Phase 1: $v = 0 + 2.5(8) = 20$ m/s, $\Delta x = \tfrac12(2.5)(64) = 80$ m. Phase 2: $0 = 20^2 + 2a(40) \Rightarrow a = -5.0\ \text{m/s}^2$; time $t = \frac{0-20}{-5} = 4.0$ s. Total time $12.0$ s, total distance $120$ m.
Common mistakes
- Using $x = vt$ when the acceleration is not zero.
- Plugging speed for $v_0$ with the wrong sign after choosing a positive direction.
- Treating average velocity as the average of speeds when acceleration is not constant.
Exam tipWrite the knowns as a list (x₀, v₀, v, a, t) with signs, mark the unknown, then choose the one equation without the quantity you do not care about.
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