Figluma mechanics worksheet

Projectile launched from a cliff

A ball is launched from a 20 m high cliff with speed 18 m s^-1 at 35 degrees above the horizontal. Ignore air resistance. Find the time of flight, horizontal range from the base of the cliff, and impact speed just before it hits the ground.

MechanicsProjectile motionKinematicsVectors

Diagram State

Projectile launched from a cliff with velocity components, downward acceleration, horizontal range, and impact velocity.

cliff
projectile path
initial velocity
horizontal velocity component
vertical velocity component
gravity
range
impact velocity
coordinate axes

Givens

Symbol	Quantity	Value	Unit
$v_{0}$	Launch speed	$18$	m s^-1
$θ$	Launch angle above horizontal	$35$	deg
$y_{0}$	Launch height	$20$	m
$g$	Gravitational field strength	$9.8$	m s^-2

Unknowns

Symbol	Quantity	Value	Unit
$t$	Time of flight	$?$	s
$R$	Horizontal range from cliff base	$?$	m
$∣ v ∣$	Impact speed	$?$	m s^-1

Coordinate System / Sign Convention

+x is horizontal to the right. +y is upward, with launch point y = 20 m and ground at y = 0.

Assumptions / Constraints Checklist

Air resistance is negligible, so horizontal acceleration is zero.
The only vertical acceleration is gravity, a_y = -g.
The launch speed and angle are measured at the instant the ball leaves the cliff.
The impact point is on level ground 20 m below the launch point.
Horizontal and vertical motion share the same time t but have different acceleration.
Horizontal velocity is constant: v_x = v_0 cos(theta).
Vertical position uses the signed displacement to the ground: 0 = y_0 + v_0 sin(theta)t - 1/2 gt^2.
Impact speed must combine horizontal and vertical velocity components.

Student Solve Checklist

Mark each row only after your setup matches the diagram state; worked equations stay in the teacher key.

SetupAxes, sign convention, model constraints, and linked-motion/origin choices are stated.
Choose projectile axes and origin
ComponentsResolved components, force directions, normal/friction setup, or velocity split are correct.
Resolve the launch velocity
Net-force / governing equationThe main Newton's law or motion equation uses the right model, signs, and shared variables.
Write the vertical position equation
ResultThe final rearrangement, numeric value, units, and direction/speed interpretation are correct.
Solve the quadratic for flight timeUse horizontal motion for rangeCombine impact velocity components

Student Working Area

Solution / Answer Key

1. Choose projectile axes and origin
$+x = right; +y = upward; y_0 = 20 m$
Use independent horizontal and vertical axes. The launch point starts above the ground, so the vertical position equation must carry y_0.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Choose projectile axes and origin" the right next equation?Listen for: Use independent horizontal and vertical axes. The launch point starts above the ground, so the vertical position equation must carry y_0.Flag if: Making gravity positive in the +y upward convention.
2. Resolve the launch velocity
$v_{x} = v_{0} cos (θ) v_{y 0} = v_{0} sin (θ)$
The horizontal component stays constant, while the vertical component changes under gravity.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Resolve the launch velocity" the right next equation?Listen for: The horizontal component stays constant, while the vertical component changes under gravity.Flag if: Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).
3. Write the vertical position equation
$0 = y_{0} + v_{0} sin (θ) t - 1/2 g t^{2}$
At impact the ball is at ground level, y = 0. Gravity is negative because +y is upward.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Write the vertical position equation" the right next equation?Listen for: At impact the ball is at ground level, y = 0. Gravity is negative because +y is upward.Flag if: Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.; Solving only for time to the top of the path, v_0 sin(theta) / g, instead of time to the ground.
4. Solve the quadratic for flight time
$t = 3.3 s$
The positive root gives the time when the projectile reaches the ground. The negative root is an extrapolated time before launch.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Solve the quadratic for flight time" the right next equation?Listen for: The positive root gives the time when the projectile reaches the ground. The negative root is an extrapolated time before launch.Flag if: Student can only quote "t = 3.3 s" without connecting it to the diagram state or givens.
5. Use horizontal motion for range
$R = v_{0} cos (θ) t = 49 m$
The range uses constant horizontal velocity multiplied by the same flight time from the vertical equation.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Use horizontal motion for range" the right next equation?Listen for: The range uses constant horizontal velocity multiplied by the same flight time from the vertical equation.Flag if: Student can only quote "R = v_0 cos(theta)t = 49 m" without connecting it to the diagram state or givens.
6. Combine impact velocity components
$∣ v ∣ = s q r t (v_{x}^{2} + v_{y}^{2}) = 27 m s^{- 1}$
Just before impact, v_x is unchanged and v_y = v_0 sin(theta) - gt. Speed is the magnitude of the velocity vector.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Combine impact velocity components" the right next equation?Listen for: Just before impact, v_x is unchanged and v_y = v_0 sin(theta) - gt. Speed is the magnitude of the velocity vector.Flag if: Reporting the vertical impact speed alone instead of combining v_x and v_y.

Diagnostic Checklist

Key checkpoint equations

1. Choose projectile axes and origin
$+x = right; +y = upward; y_0 = 20 m$
2. Resolve the launch velocity
$v_{x} = v_{0} cos (θ) v_{y 0} = v_{0} sin (θ)$
3. Write the vertical position equation
$0 = y_{0} + v_{0} sin (θ) t - 1/2 g t^{2}$
4. Solve the quadratic for flight time
$t = 3.3 s$
5. Use horizontal motion for range
$R = v_{0} cos (θ) t = 49 m$
6. Combine impact velocity components
$∣ v ∣ = s q r t (v_{x}^{2} + v_{y}^{2}) = 27 m s^{- 1}$

Common Wrong Paths

Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).
Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.
Solving only for time to the top of the path, v_0 sin(theta) / g, instead of time to the ground.
Dropping the initial height term from the vertical displacement equation.
Making gravity positive in the +y upward convention.
Reporting the vertical impact speed alone instead of combining v_x and v_y.

Wrong Answer Signals

Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).Usually indicates the "Resolve the launch velocity" checkpoint needs review.
$v_{x} = v_{0} cos (θ) v_{y 0} = v_{0} sin (θ)$
Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.Usually indicates the "Write the vertical position equation" checkpoint needs review.
$0 = y_{0} + v_{0} sin (θ) t - 1/2 g t^{2}$
Solving only for time to the top of the path, v_0 sin(theta) / g, instead of time to the ground.Usually indicates the "Write the vertical position equation" checkpoint needs review.
$0 = y_{0} + v_{0} sin (θ) t - 1/2 g t^{2}$
Dropping the initial height term from the vertical displacement equation.Usually indicates the "Write the vertical position equation" checkpoint needs review.
$0 = y_{0} + v_{0} sin (θ) t - 1/2 g t^{2}$
Making gravity positive in the +y upward convention.Usually indicates the "Choose projectile axes and origin" checkpoint needs review.
$+x = right; +y = upward; y_0 = 20 m$
Reporting the vertical impact speed alone instead of combining v_x and v_y.Usually indicates the "Combine impact velocity components" checkpoint needs review.
$∣ v ∣ = s q r t (v_{x}^{2} + v_{y}^{2}) = 27 m s^{- 1}$

Tutor Marking Rubric

Tutor score rows use curated Figluma checkpoints as marking cues. They are not automated grading or a symbolic mark scheme.

Tutor Mark Sheet

Manual tutor mark sheet only. Use observed work and leave rows blank when evidence is copied from a reveal.

Total: ___ / 8

Setup

Choose projectile axes and origin

___ / 2

Axes, sign convention, model constraints, and linked-motion/origin choices are stated.If weak, reteach the row from this signal: Making gravity positive in the +y upward convention.

Score descriptions

0No usable evidence for this row, or the work contradicts "Choose projectile axes and origin".
1Partly correct, but review this row's checkpoint signal: Making gravity positive in the +y upward convention.
2Complete row: Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

Watch signals

Making gravity positive in the +y upward convention.

Components

Resolve the launch velocity

___ / 2

Resolved components, force directions, normal/friction setup, or velocity split are correct.If weak, reteach the row from this signal: Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).

Score descriptions

0No usable evidence for this row, or the work contradicts "Resolve the launch velocity".
1Partly correct, but review this row's checkpoint signal: Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).
2Complete row: Resolved components, force directions, normal/friction setup, or velocity split are correct.

Watch signals

Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).

Net-force / governing equation

Write the vertical position equation

___ / 2

The main Newton's law or motion equation uses the right model, signs, and shared variables.If weak, reteach the row from this signal: Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.

Score descriptions

0No usable evidence for this row, or the work contradicts "Write the vertical position equation".
1Partly correct, but review this row's checkpoint signal: Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.
2Complete row: The main Newton's law or motion equation uses the right model, signs, and shared variables.

Watch signals

Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.
Solving only for time to the top of the path, v_0 sin(theta) / g, instead of time to the ground.

Result

Solve the quadratic for flight time; Use horizontal motion for range; Combine impact velocity components

___ / 2

The final rearrangement, numeric value, units, and direction/speed interpretation are correct.If weak, reteach the row from this signal: Reporting the vertical impact speed alone instead of combining v_x and v_y.

Score descriptions

0No usable evidence for this row, or the work contradicts "Solve the quadratic for flight time".
1Partly correct, but review this row's checkpoint signal: Reporting the vertical impact speed alone instead of combining v_x and v_y.
2Complete row: The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

Watch signals

Reporting the vertical impact speed alone instead of combining v_x and v_y.

Total: ___ / 8Add the four row scores after checking actual student evidence. Leave it blank when the work is incomplete or only copied from reveals.Reteach from the first 0 or 1 row before assigning another variant or adding a new problem.

Setup

Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

012

Score guide

0No usable evidence for this row, or the work contradicts "Choose projectile axes and origin".
1Partly correct, but review this row's checkpoint signal: Making gravity positive in the +y upward convention.
2Complete row: Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

Checkpoints

Choose projectile axes and origin
$+x = right; +y = upward; y_0 = 20 m$

Watch for

Making gravity positive in the +y upward convention.

Components

Resolved components, force directions, normal/friction setup, or velocity split are correct.

012

Score guide

0No usable evidence for this row, or the work contradicts "Resolve the launch velocity".
1Partly correct, but review this row's checkpoint signal: Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).
2Complete row: Resolved components, force directions, normal/friction setup, or velocity split are correct.

Checkpoints

Resolve the launch velocity
$v_{x} = v_{0} cos (θ) v_{y 0} = v_{0} sin (θ)$

Watch for

Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).

Net-force / governing equation

The main Newton's law or motion equation uses the right model, signs, and shared variables.

012

Score guide

0No usable evidence for this row, or the work contradicts "Write the vertical position equation".
1Partly correct, but review this row's checkpoint signal: Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.
2Complete row: The main Newton's law or motion equation uses the right model, signs, and shared variables.

Checkpoints

Write the vertical position equation
$0 = y_{0} + v_{0} sin (θ) t - 1/2 g t^{2}$

Watch for

Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.
Solving only for time to the top of the path, v_0 sin(theta) / g, instead of time to the ground.
Dropping the initial height term from the vertical displacement equation.

Result

The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

012

Score guide

0No usable evidence for this row, or the work contradicts "Solve the quadratic for flight time".
1Partly correct, but review this row's checkpoint signal: Reporting the vertical impact speed alone instead of combining v_x and v_y.
2Complete row: The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

Checkpoints

Solve the quadratic for flight time
$t = 3.3 s$
Use horizontal motion for range
$R = v_{0} cos (θ) t = 49 m$
Combine impact velocity components
$∣ v ∣ = s q r t (v_{x}^{2} + v_{y}^{2}) = 27 m s^{- 1}$

Watch for

Reporting the vertical impact speed alone instead of combining v_x and v_y.

Diagram State

Givens

Unknowns

Coordinate System / Sign Convention

Assumptions / Constraints Checklist

Student Solve Checklist

Student Working Area

1. Choose projectile axes and origin

2. Resolve the launch velocity

3. Write the vertical position equation

4. Solve the quadratic for flight time

5. Use horizontal motion for range

6. Combine impact velocity components

Diagnostic Checklist

Key checkpoint equations

Common Wrong Paths

Wrong Answer Signals

Tutor Marking Rubric

Tutor Mark Sheet

Setup

Components

Net-force / governing equation

Result

Setup

Score guide

Checkpoints

Watch for

Components

Score guide

Checkpoints

Watch for

Net-force / governing equation

Score guide

Checkpoints

Watch for

Result

Score guide

Checkpoints

Watch for