INTERACTIVE LEARNING LAB
Projectile Motion Calculator
Launch a projectile, break its motion apart, follow it through time, and build toward solving problems independently.
New to projectile motion? Start with Guided Mode. Already know the physics? Jump to the Quick Calculator.
STAGE 1
Launch Your Projectile
Set the speed, angle, and gravity — watch the trajectory appear
YOUR PROJECTILE
Set the launch conditions. The trajectory updates live.
SPEED + ANGLE + GRAVITY → TRAJECTORY
LIVE ARC · SCROLL TO ZOOM · DOUBLE-CLICK TO RESET
STAGE 2
Break the Velocity Apart
The launch velocity splits into horizontal and vertical components
The initial velocity u points along the launch angle. We split it into:ux = u cos(θ) · uy = u sin(θ)
WHAT JUST HAPPENED?
The single launch velocity becomes two independent pieces: a horizontal speed that gravity does not touch, and a vertical speed that gravity will steadily reduce.
STAGE 3
One Projectile — Two Motions
Horizontal and vertical motion are independent, but share the same time
HORIZONTAL MOTION
ax = 0
vx = ux = 17.7 m/s
x = ux · t
No horizontal acceleration — velocity stays constant.
VERTICAL MOTION
ay = −g = −9.81
vy = uy − gt
y = y₀ + uy·t − ½gt²
Gravity steadily reduces the upward velocity.
Time connects the two motions. The horizontal and vertical motions can be analyzed separately, but they describe the same projectile at the same instant. Both use the same t.
FOLLOW THE FLIGHT
WHAT JUST HAPPENED?
Drag the slider to watch the projectile travel. The horizontal velocity never changes, while the vertical velocity decreases on the way up, hits zero at the peak, and grows downward on the way down.
STAGE 4
Reach the Peak
What is special about the velocity at maximum height?
What is special about the velocity at maximum height?
Misconception check: At maximum height, is the projectile stationary?No — unless launched straight up. vy = 0 but vx remains, so total velocity and kinetic energy are not zero.
WHAT JUST HAPPENED?
The projectile has reached its highest point. Gravity reduced its upward vertical velocity to zero, but its horizontal velocity is unchanged. It continues forward while beginning to fall.
STAGE 5
Find Position at a Distance
Use horizontal motion to find time, then use that time vertically
WHERE IS THE PROJECTILE HERE?
STAGE 5
Find the Landing Point
Where does the projectile intersect the landing level?
We solve y_target = y₀ + uy·t − ½gt² for the physically meaningful nonnegative time, then compute R = ux × t_flight.
WHAT JUST HAPPENED?
The projectile lands when its vertical position returns to the landing level. The larger time solution of the quadratic gives the flight time; multiplying by the constant horizontal speed gives the range.
STAGE 6
Follow the Velocity
Find speed and direction at any point along the trajectory
VELOCITY AT ANY TIME
ADVANCED: VELOCITY AT A SPECIFIED HEIGHT
A projectile can pass the same height twice — once rising, once falling.
WHAT JUST HAPPENED?
At any time, the horizontal velocity stays the same while the vertical velocity changes. The direction angle tells you whether the projectile is rising (positive) or falling (negative).
STAGE 7
Track the Energy
Kinetic and potential energy exchange while total stays constant
KE
313 J
GPE
0 J
TOTAL
313 J
Total mechanical energy is constant in the ideal model (no air resistance). As the projectile rises, KE decreases and GPE increases. As it falls, KE increases and GPE decreases.
WHAT JUST HAPPENED?
Energy shifts between kinetic and gravitational potential, but the total stays the same (conservation of mechanical energy). The reference level for GPE is y = 0 — choosing a different reference changes GPE but not the physics.
STAGE 8
Master Projectile Motion
You've worked through the full guided sequence
0 of 8 stages completed.
Ready to try this without help? Switch to Practice mode for original problems with gradually reduced assistance — from fully guided hints to independent problem-solving.
LIVE TRAJECTORY
ADVANCED EXPLORATION
Experiment with the Physics
Keep speed and gravity constant. Drag the angle and watch the trajectory change.
Angle: 45°
For same-height launch without air resistance, 45° gives maximum range. Complementary angles (e.g. 30° and 60°) give the same range.
CONCEPT CHECKS
Test Your Understanding
Which velocity component changes because of gravity?
What is vy at maximum height?
Does vx become zero at maximum height?
Can the projectile have the same height at two different times?
At maximum height, is the projectile stationary?
COMMON MISTAKES
Projectile Motion Pitfalls
Using total velocity instead of a component
Always split u into ux and uy before solving. The horizontal equation uses ux, not u.
Swapping sin and cos
Adjacent side = cos, opposite side = sin. ux (horizontal) = u cos θ, uy (vertical) = u sin θ.
Assuming horizontal velocity changes
Gravity acts vertically. In ideal motion, vx is constant — it never changes.
Thinking velocity is zero at max height
Only vy = 0 at the peak. vx remains nonzero (unless launched straight up), so the projectile is still moving.
Using the range shortcut at different heights
R = u²sin(2θ)/g only works when launch and landing heights are equal. Otherwise use the quadratic.
Wrong sign for gravity
g is a positive magnitude. The vertical acceleration is ay = −g. The −½gt² term subtracts.
Confusing distance with displacement
Displacement is a vector (start to end). Distance is the path length. They differ for projectiles.
Forgetting two times at one height
A height below the peak is reached twice — rising and falling. Don't discard the second solution.
Degrees vs radians
Make sure your calculator is in degree mode when the angle is given in degrees.
Treating x and y as unrelated times
Horizontal and vertical motion share the same time t. Never use different times for the same instant.
QUICK REFERENCE
Equation Summary
COMPONENTS
ux = u cos θ
uy = u sin θ
POSITION
x = x₀ + ux t
y = y₀ + uy t − ½gt²
VELOCITY
vx = ux (constant)
vy = uy − gt
v = √(vx² + vy²)
PEAK
vy = 0 at peak
t_top = uy / g
H = y₀ + uy² / (2g)
RANGE (SAME HEIGHT)
T = 2uy / g
R = ux T = u² sin(2θ) / g
ENERGY
KE = ½mv²
GPE = mgy
E = KE + GPE = constant
Sign convention: +x right, +y up, g positive magnitude, ay = −g. All equations assume ideal motion (no air resistance).
Sign convention: +x right, +y up, g entered as positive magnitude, ay = −g. All physics uses established real-world projectile-motion equations. Fictional scenarios may use different gravity values but the equations remain scientifically consistent.
Explore the Real Science
The physics behind every launch — motion, energy, and waves — is real. Explore it hands-on.
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