[繁體中文] Projectile Motion Practice Questions: 10 Problems With Full Solutions
[繁體中文] Projectile motion practice questions are the fastest way to get comfortable with trajectories because you work through real numbers — not just read the formulas. This set of 10 practice questions moves from easy warm-ups to challenge-level problems, covering horizontal launches, angled launches, and launches from a height. Every problem uses g = 9.8 m/s², shows the full arithmetic, and ends with a final answer you can check your own work against, plus a section on the mistakes that cost students the most points and how to verify any answer without a calculator app.
目錄
- 01[繁體中文] Why Practice Projectile Motion Problems Instead of Just Reviewing the Formulas?
- 02[繁體中文] What Formulas Do You Need Before Starting These Practice Questions?
- 03[繁體中文] 3 Easy Warm-Up Questions (With Full Solutions)
- 04[繁體中文] How Do You Solve a Horizontal Launch Practice Problem at the Medium Level?
- 05[繁體中文] How Do You Solve an Angled Launch Practice Problem?
- 06[繁體中文] What Changes When the Launch Point Is Above the Landing Point?
- 07[繁體中文] 4 Challenge-Level Practice Questions to Test Yourself
- 08[繁體中文] What Mistakes Cost the Most Points on Projectile Motion Practice Questions?
- 09[繁體中文] How Can You Verify Your Own Projectile Motion Answers Are Correct?
- 10[繁體中文] Ready to Check Your Own Projectile Motion Work?
[繁體中文] Why Practice Projectile Motion Problems Instead of Just Reviewing the Formulas?
[繁體中文] Reading through the projectile motion equations feels like understanding, but it is not the same skill as solving a problem cold — recognizing which formula fits which setup, keeping components separate, and catching your own algebra mistakes before they turn into a wrong final answer. Projectile motion trips students up less because the formulas are hard and more because there are several of them, and picking the right one depends on details easy to skim past: is the launch height equal to the landing height, does the object start with any vertical velocity, is the question asking for a value above the launch point or above the ground? Practice questions force you to notice those details every time, which is exactly the skill tested on quizzes and exams. The 10 questions below are grouped into easy, medium, and challenge levels. Work each one on paper before reading the solution — even getting partway through and comparing your setup to the worked steps builds far more understanding than reading a finished solution passively. Every problem uses g = 9.8 m/s² and shows the arithmetic in full, so you can find exactly where your own work diverges if your answer does not match.
[繁體中文] The skill that separates a passing projectile motion score from a struggling one is not the formulas themselves — it is recognizing which formula a given setup calls for.
[繁體中文] What Formulas Do You Need Before Starting These Practice Questions?
[繁體中文] Four equations cover essentially every projectile motion practice question you will encounter. Keep these in front of you while you work through the problems below, and note the one condition attached to the range shortcut — it is the single most common source of wrong answers.
1. [繁體中文] Velocity components
[繁體中文] vₓ₀ = v₀ × cos(θ) and v_y₀ = v₀ × sin(θ), where v₀ is launch speed and θ is the launch angle above horizontal. For a purely horizontal launch, θ = 0°, so vₓ₀ = v₀ and v_y₀ = 0.
2. [繁體中文] Time of flight
[繁體中文] When launch and landing heights are equal: T = 2v_y₀ ÷ g. When the launch point is above or below the landing point by a height h, solve h + v_y₀t − ½gt² = 0 with the quadratic formula instead — the equal-height shortcut does not apply.
3. [繁體中文] Maximum height
[繁體中文] H = v_y₀² ÷ (2g), measured above the launch point. Add the launch height if the question asks for height above the ground instead.
4. [繁體中文] Range
[繁體中文] R = vₓ₀ × T always works, using whatever time of flight you found above. The shortcut R = v₀² × sin(2θ) ÷ g only works when launch height equals landing height — using it for a launch from a height is the fastest way to get a wrong answer.
[繁體中文] R = vₓ₀ × T works for every projectile motion question. R = v₀²sin(2θ)/g only works when the launch and landing heights match — treat it as a shortcut, not a rule.
[繁體中文] 3 Easy Warm-Up Questions (With Full Solutions)
[繁體中文] These three questions use simple numbers and straightforward setups — no quadratic formula required. Try each one before checking the worked solution beneath it.
1. [繁體中文] Q1 (Easy) — A book slides off a table
[繁體中文] A book slides off a horizontal table 0.72 m high with a speed of 1.8 m/s. How long does it take to hit the floor, and how far from the table does it land? Solution: since the launch is horizontal, v_y₀ = 0, so t = √(2h ÷ g) = √(2 × 0.72 ÷ 9.8) = √(1.44 ÷ 9.8) = √0.1469 ≈ 0.38 s. Range: R = vₓ₀ × t = 1.8 × 0.38 ≈ 0.69 m. Answer: about 0.38 s and 0.69 m.
2. [繁體中文] Q2 (Easy) — Maximum height only
[繁體中文] A ball is launched from ground level at 8 m/s at 50° above horizontal. Find its maximum height. Solution: v_y₀ = 8 × sin(50°) = 8 × 0.766 ≈ 6.13 m/s. H = v_y₀² ÷ (2g) = 6.13² ÷ 19.6 = 37.58 ÷ 19.6 ≈ 1.92 m. Answer: about 1.92 m.
3. [繁體中文] Q3 (Easy) — A dart thrown horizontally
[繁體中文] A dart is thrown horizontally at 5 m/s from a height of 1.25 m. Find the range. Solution: t = √(2 × 1.25 ÷ 9.8) = √(2.5 ÷ 9.8) = √0.2551 ≈ 0.51 s. R = 5 × 0.51 ≈ 2.53 m. Answer: about 2.53 m.
[繁體中文] Every horizontal-launch problem collapses to the same shortcut: t = √(2h/g). Recognizing v_y₀ = 0 immediately is the fastest way through an easy-level question.
[繁體中文] How Do You Solve a Horizontal Launch Practice Problem at the Medium Level?
[繁體中文] Medium-level horizontal launches use the same method as the easy questions but with larger, less convenient numbers — good practice for keeping the arithmetic accurate under exam pressure.
1. [繁體中文] Q4 (Medium) — A golf ball off a cliff
[繁體中文] A golf ball rolls off the edge of a 22 m cliff at a horizontal speed of 14 m/s. Find the time of flight and the range.
2. [繁體中文] Step 1 — Time of flight
[繁體中文] v_y₀ = 0, so t = √(2h ÷ g) = √(2 × 22 ÷ 9.8) = √(44 ÷ 9.8) = √4.490 ≈ 2.12 s.
3. [繁體中文] Step 2 — Range
[繁體中文] R = vₓ₀ × t = 14 × 2.12 ≈ 29.7 m.
4. [繁體中文] Step 3 — Verify
[繁體中文] A 22 m cliff (about a 7-story drop) taking just over 2 seconds to reach the ground matches free-fall intuition, and 14 m/s carrying the ball roughly 30 m in that time is a reasonable, non-extreme range. Answer: t ≈ 2.12 s, R ≈ 29.7 m.
[繁體中文] Larger numbers do not change the method — they only make it easier to lose a decimal point. Write out every intermediate value instead of chaining calculator operations.
[繁體中文] How Do You Solve an Angled Launch Practice Problem?
[繁體中文] Angled launches from ground level are the most common projectile motion question type, and this example works through all three standard outputs — maximum height, time of flight, and range — from a single launch.
1. [繁體中文] Q5 (Medium) — A javelin throw
[繁體中文] A javelin is launched from ground level at 22 m/s at 42° above horizontal. Find the maximum height, the time of flight, and the range.
2. [繁體中文] Step 1 — Components
[繁體中文] vₓ₀ = 22 × cos(42°) = 22 × 0.7431 ≈ 16.35 m/s. v_y₀ = 22 × sin(42°) = 22 × 0.6691 ≈ 14.72 m/s.
3. [繁體中文] Step 2 — Maximum height
[繁體中文] H = v_y₀² ÷ (2g) = 14.72² ÷ 19.6 = 216.7 ÷ 19.6 ≈ 11.06 m.
4. [繁體中文] Step 3 — Time of flight
[繁體中文] Launch and landing heights are equal (both ground level), so T = 2v_y₀ ÷ g = 2 × 14.72 ÷ 9.8 = 29.44 ÷ 9.8 ≈ 3.00 s.
5. [繁體中文] Step 4 — Range
[繁體中文] R = vₓ₀ × T = 16.35 × 3.00 ≈ 49.1 m. Cross-check with the shortcut since heights are equal: R = v₀² × sin(2θ) ÷ g = 484 × sin(84°) ÷ 9.8 = 484 × 0.9945 ÷ 9.8 ≈ 49.1 m — the two methods match. Answer: H ≈ 11.06 m, T ≈ 3.00 s, R ≈ 49.1 m.
[繁體中文] When launch and landing heights are equal, R = vₓ₀ × T and the shortcut formula should always land within rounding error of each other — a free built-in check.
[繁體中文] What Changes When the Launch Point Is Above the Landing Point?
[繁體中文] Launch-from-a-height problems are the challenge-level staple because the equal-height time shortcut no longer applies — you have to fall back on the full quadratic equation.
1. [繁體中文] Q6 (Challenge) — A cannonball from a wall
[繁體中文] A cannonball is fired from the top of a 30 m wall at 25 m/s at 30° above horizontal. Find the time of flight and the range.
2. [繁體中文] Step 1 — Components
[繁體中文] vₓ₀ = 25 × cos(30°) = 25 × 0.8660 ≈ 21.65 m/s. v_y₀ = 25 × sin(30°) = 25 × 0.5 = 12.5 m/s.
3. [繁體中文] Step 2 — Set up the vertical equation
[繁體中文] Measuring downward displacement as −30 m relative to the launch point: −30 = 12.5t − 4.9t², which rearranges to 4.9t² − 12.5t − 30 = 0.
4. [繁體中文] Step 3 — Solve with the quadratic formula
[繁體中文] t = [12.5 ± √(12.5² + 4 × 4.9 × 30)] ÷ (2 × 4.9) = [12.5 ± √(156.25 + 588)] ÷ 9.8 = [12.5 ± √744.25] ÷ 9.8 = [12.5 ± 27.28] ÷ 9.8. The negative root is not physical, so t = (12.5 + 27.28) ÷ 9.8 ≈ 4.06 s.
5. [繁體中文] Step 4 — Range and verification
[繁體中文] R = vₓ₀ × t = 21.65 × 4.06 ≈ 87.9 m. Verify by plugging t back into the vertical equation: 12.5 × 4.06 − 4.9 × 4.06² = 50.75 − 80.77 ≈ −30.0 m, which matches the wall height with the correct sign. Answer: t ≈ 4.06 s, R ≈ 87.9 m.
[繁體中文] Whenever launch and landing heights differ, skip every shortcut formula and go straight to h + v_y₀t − ½gt² = 0 solved with the quadratic formula — it is the one method that never fails.
[繁體中文] 4 Challenge-Level Practice Questions to Test Yourself
[繁體中文] These four questions combine an angled launch with a height difference, the hardest standard combination. Work through the full method — components, the quadratic equation for time, then range — and check your final numbers against the answers below.
1. [繁體中文] Q7 (Challenge) — Trebuchet from a hill
[繁體中文] A trebuchet launches a stone at 30 m/s at 55° above horizontal from a hill 15 m above the target field. Find the range. Answer: t ≈ 5.57 s, R ≈ 95.8 m.
2. [繁體中文] Q8 (Easy–Medium) — Frisbee off a ledge
[繁體中文] A frisbee is thrown horizontally at 9 m/s from a height of 1.5 m. Find the time of flight and range. Answer: t ≈ 0.55 s, R ≈ 4.98 m.
3. [繁體中文] Q9 (Medium) — Ground-level angled launch
[繁體中文] A ball is launched from ground level at 18 m/s at 25° above horizontal. Find the maximum height, time of flight, and range. Answer: H ≈ 2.95 m, T ≈ 1.55 s, R ≈ 25.3 m.
4. [繁體中文] Q10 (Challenge) — Platform launch at a steep angle
[繁體中文] A ball is launched at 40 m/s at 60° above horizontal from a platform 10 m high. Find the time of flight and the range. Answer: t ≈ 7.35 s, R ≈ 146.9 m.
[繁體中文] If your answer to a challenge question is off by more than rounding error, redo the quadratic formula step first — a sign error inside the square root is the most common place these problems go wrong.
[繁體中文] What Mistakes Cost the Most Points on Projectile Motion Practice Questions?
[繁體中文] The same handful of errors account for most lost points on projectile motion problems, and every one of them is avoidable by slowing down at a specific step.
1. [繁體中文] Using the range shortcut when heights are unequal
[繁體中文] R = v₀²sin(2θ)/g only holds when the launch and landing heights are the same. Applying it to a launch from a cliff or platform gives a confidently wrong answer — always default to R = vₓ₀ × T unless you have explicitly checked the heights match.
2. [繁體中文] Dropping the sign on height in the quadratic
[繁體中文] When solving h + v_y₀t − ½gt² = 0 for a launch above the landing point, h should be entered as a negative displacement relative to the launch point, or the equation should be set up as −h = v_y₀t − ½gt². Mixing the sign convention flips the answer entirely.
3. [繁體中文] Mixing degrees and the wrong trig function
[繁體中文] Swapping sine and cosine — using cos(θ) for the vertical component instead of sine — is the single most common component-decomposition error. Double-check that the vertical component uses sin(θ) and the horizontal uses cos(θ) before plugging numbers in.
4. [繁體中文] Rounding intermediate values too early
[繁體中文] Rounding vₓ₀ or v_y₀ to two significant figures before using them in later steps compounds error through the rest of the problem. Carry at least four significant figures through intermediate steps and round only the final answer.
[繁體中文] Most projectile motion mistakes are not conceptual — they are a shortcut formula applied outside the one condition it requires.
[繁體中文] How Can You Verify Your Own Projectile Motion Answers Are Correct?
[繁體中文] Three checks catch nearly every projectile motion mistake before you turn in an answer, and none of them require a calculator app.
1. [繁體中文] Units check
[繁體中文] Track units through every formula. √(m ÷ (m/s²)) should simplify to seconds; (m/s)² ÷ (m/s²) should simplify to meters. If the units do not cancel cleanly, a term was mishandled somewhere in the algebra.
2. [繁體中文] Magnitude check
[繁體中文] Ask whether the number makes sense for the scenario. A ball dropped from table height should take well under a second to land; a cannonball launched at tens of meters per second should travel tens to hundreds of meters, not thousands. An answer that is off by an order of magnitude usually means a squared term got dropped.
3. [繁體中文] Plug-back check
[繁體中文] Substitute your computed time back into the original vertical position equation and confirm it returns the height you started with, as in Q6 above. This single step catches sign errors and quadratic-formula mistakes that a units or magnitude check might miss.
[繁體中文] Plugging your answer back into the equation you started from is the single most reliable check — it catches the errors that units and magnitude checks both miss.
[繁體中文] Ready to Check Your Own Projectile Motion Work?
[繁體中文] Working through practice questions by hand is what builds real understanding, but knowing where your own solution went wrong is just as valuable as getting the right answer. Solvify's Smart Scan Solver lets you photograph a projectile motion problem — your own homework, a worksheet, or a question from this list — and see a full step-by-step solution to compare against your work, so you can find exactly which line your answer diverged on. Keep working through problems by hand first; use the tool to check, not to skip the practice.
[繁體中文] The goal of practice questions is not the final number — it is being able to set up the next problem correctly on your own.
相關文章
[繁體中文] Projectile Motion Calculator: How to Solve Any Trajectory by Hand
[繁體中文] The full equation set and two worked examples behind every projectile motion calculation, if you need a formula refresher before these practice questions.
[繁體中文] Physics Problem Solving: A Step-by-Step Method That Works
[繁體中文] A general five-step framework for setting up any physics problem, useful for organizing projectile motion questions before you start computing.
[繁體中文] How to Calculate Acceleration From a Velocity-Time Graph
[繁體中文] Practice reading constant acceleration from graphs, the same g = 9.8 m/s² relationship that drives every vertical projectile motion equation.
相關數學解題工具
[繁體中文] Practice Mode
[繁體中文] Work through additional projectile motion questions with instant feedback on each step, not just the final answer.
[繁體中文] Smart Scan Solver
[繁體中文] Snap a photo of any projectile motion practice question and get a full step-by-step solution to check your work against.
[繁體中文] Step-by-Step Solutions
[繁體中文] See every intermediate value — vₓ₀, v_y₀, time of flight — not just the final range or height.
相關學科
[繁體中文] Physics Help
[繁體中文] Kinematics, forces, and energy problems solved with the same structured, verifiable method used throughout this guide.
[繁體中文] Algebra Help
[繁體中文] The equation-solving fluency behind isolating time, height, or range from any projectile motion formula, including the quadratic cases.
[繁體中文] Geometry Help
[繁體中文] Angle and triangle reasoning that underlies decomposing a launch velocity into horizontal and vertical components.