PHSX 220 Physics I (w/ Calculus) · Unit 1: Kinematics & Newton’s laws · on Exam 1
Ch 3.5–3.6 Free fall
Study notes for Free fall 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
- Near Earth's surface, ignoring air resistance, every object accelerates downward at $g = 9.8\ \text{m/s}^2$ regardless of mass. Choose up as positive and use $a = -g$ throughout the flight, including at the top.
- At the highest point the velocity is zero but the acceleration is still $-g$. The time up equals the time down for a launch-and-return to the same height, and the return speed equals the launch speed.
- A dropped object: $v = -gt$, $y = y_0 - \tfrac12 gt^2$, impact speed $\sqrt{2gh}$. Thrown up at $v_0$: max height $v_0^2/(2g)$, time to top $v_0/g$.
- When the landing height differs from the launch height, use $y = y_0 + v_0 t - \tfrac12 g t^2$ and solve the quadratic; keep the positive root.
Key formulas
Free fall (up positive)
$$v = v_0 - gt,\quad y = y_0 + v_0t - \tfrac12 gt^2,\quad v^2 = v_0^2 - 2g\,\Delta y$$
Handy results
$$t_{\text{drop}} = \sqrt{2h/g},\quad v_{\text{impact}} = \sqrt{2gh},\quad h_{\max} = \frac{v_0^2}{2g},\quad t_{\text{top}} = \frac{v_0}{g}$$
Worked example
A ball is thrown straight up at 15 m/s from the edge of a 20 m building. Find the maximum height above the ground, the time to hit the ground, and the impact speed.
Show the solution
Max height: $20 + \frac{15^2}{2(9.8)} = 20 + 11.5 = 31.5$ m. Landing: $-20 = 15t - 4.9t^2 \Rightarrow 4.9t^2 - 15t - 20 = 0 \Rightarrow t = \frac{15 + \sqrt{225 + 392}}{9.8} = 4.07$ s. Impact speed: $v^2 = 15^2 + 2(9.8)(20) = 617 \Rightarrow v = 24.8$ m/s.
Common mistakes
- Setting a = 0 at the top of the flight.
- Forgetting that a downward displacement is negative when up is positive.
- Using g = 10 on a problem set that specifies 9.8 (Expert TA tolerance is tight).
Exam tipSketch the trajectory with y = 0 at the ground and label y₀, v₀ and the final y. Half the free-fall mistakes are sign mistakes.
Keep going in Mathub
Mathub is free for Montana State students: an endless quizzer with worked explanations, one-question lessons, flashcards, exam prep checklists, the class calendar with every deadline, a grade calculator and a study board for PHSX 220.