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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.

State

Givens

$v_{0}$ Launch speed: $18 m s^{- 1}$
$θ$ Launch angle above horizontal: $3 5^{\circ}$
$y_{0}$ Launch height: $20 m$
$g$ Gravitational field strength: $9.8 m s^{- 2}$

Unknowns

$t$ Time of flight: $? s$
$R$ Horizontal range from cliff base: $? m$
$∣ v ∣$ Impact speed: $? m s^{- 1}$

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

Assumptions

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.

Constraints

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.

Projectile launched from a cliff

Active checkpoint: Choose projectile axes and origin

LookSetupChoose projectile axes and origin

ChooseChoose projectile axes and originUse independent horizontal and vertical axes. The launch point starts above the ground, so the vertical position equation must carry y_0.

AvoidWrong turnMaking gravity positive in the +y upward convention.

Equation lens

+ x

right; +y = upward; y_0 = 20 m

Choose projectile axes and originBefore revealing the worked equation, decide what "Choose projectile axes and origin" must establish from the diagram and givens.

Hint before equation

Set the axes, origin, sign convention, or linked-motion choice before writing algebra.

Making gravity positive in the +y upward convention.

Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Choose projectile axes and origin" the right next equation?

Step 1 of 6SetupChoose projectile axes and origin

Use the diagram focus, problem state, and scratch workspace first.

Checkpoints

Choose projectile axes and origin
Not yet
Set the axes, origin, sign convention, or linked-motion choice before writing algebra.
Watch: Making gravity positive in the +y upward convention.
Resolve the launch velocity
Not yet
Resolve only onto the chosen axes and check which part of each vector belongs in this step.
Watch: Using the full launch speed as the horizontal velocity instead of v_0 cos(theta).
Write the vertical position equation
Not yet
Separate driving terms from opposing terms, then assign signs from the positive direction.
Watch: Using the same-height range formula v_0^2 sin(2theta) / g even though the projectile lands below launch height.
Solve the quadratic for flight time
Not yet
Substitute after the model equation is settled, then check units and the requested interpretation.
Watch: Reporting the vertical impact speed alone instead of combining v_x and v_y.
Use horizontal motion for range
Not yet
Substitute after the model equation is settled, then check units and the requested interpretation.
Watch: Reporting the vertical impact speed alone instead of combining v_x and v_y.
Combine impact velocity components
Not yet
Substitute after the model equation is settled, then check units and the requested interpretation.
Watch: Reporting the vertical impact speed alone instead of combining v_x and v_y.

Common traps

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.

Solve Workspace

Solve checklist0 / 4 marked

Checkpoint reveal0 / 6 revealed

Checkpoint progress0 / 6 worked; 0 checked

Equation reasons0 / 6 noted

SetupAxes, sign convention, model constraints, and linked-motion/origin choices are stated.
Choose projectile axes and origin
1 common trap tied to this row.
ComponentsResolved components, force directions, normal/friction setup, or velocity split are correct.
Resolve the launch velocity
1 common trap tied to this row.
Net-force / governing equationThe main Newton's law or motion equation uses the right model, signs, and shared variables.
Write the vertical position equation
3 common traps tied to this row.
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
1 common trap tied to this row.

Why this equation?

Local note for the active checkpoint. It is copied into the attempt snapshot for validation.

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