Free tool
Enter a launch speed, angle and launch height to get the time of flight, range, maximum height and the velocity at impact - with every step of the working shown.
Built for College Physics I and AP Physics 10 for a horizontal launch, negative if thrown downward. Allowed range -90 to 90.
0 for level ground, for example 10 for a launch from a 10 m cliff.
Enter a launch speed and angle to see the trajectory.
Projectile problems get fast once the setup is automatic. Encodr keeps equations like these on a spaced-repetition schedule so they are still there on exam day.
Get started freeHorizontal and vertical motion are independent and share only the time. Split the launch velocity into
v0x = v0 cos θ and v0y = v0 sin θ. Horizontally there is no acceleration,
so x = v0x t. Vertically the acceleration is g downward, so y = h0 + v0y t - (1/2) g t².
The projectile lands when y = 0. That is a quadratic in t, and the positive root is the time of flight:
t = (v0y + √(v0y² + 2 g h0)) ÷ g. The range is v0x t. The peak comes
when the vertical velocity reaches zero, at t = v0y ÷ g, and the height gained above the
launch point is v0y² ÷ 2g. At impact the horizontal velocity is still v0x and the
vertical velocity is -√(v0y² + 2 g h0).
On level ground (h0 = 0) these reduce to the textbook shortcuts R = v0² sin 2θ ÷ g,
H = (v0 sin θ)² ÷ 2g and T = 2 v0 sin θ ÷ g. The shortcuts are
wrong when the launch and landing heights differ, which is why this tool always solves the quadratic.
A ball is launched at 20.0 m/s at 30.0° above level ground, with g = 9.80 m/s². Then v0x = 20.0 cos 30.0° = 17.3 m/s and v0y = 20.0 sin 30.0° = 10.0 m/s. Time of flight T = 2 × 10.0 ÷ 9.80 = 2.04 s, range R = 400 × sin 60.0° ÷ 9.80 = 35.3 m, and maximum height H = 10.0² ÷ 19.6 = 5.10 m. A 60.0° launch at the same speed lands at the same 35.3 m, because complementary angles give equal ranges.
This model ignores air resistance, which shortens real ranges and pulls the best angle below 45°. The step-by-step walk-through is in projectile motion equations with worked examples, and the full course is the free College Physics I deck. For one-dimensional motion, use the kinematics calculator.
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