Projectile Motion Calculator

Range, maximum height and time of flight for a projectile launched at a speed and angle, ignoring air resistance.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

A projectile moves at a constant speed horizontally and with constant acceleration (gravity) vertically. Split the launch speed into horizontal and vertical components, find the time to land from the vertical motion, then multiply the horizontal speed by that time to get the range.

The results ignore air resistance, so they are most accurate for dense, slow projectiles. For a launch from the ground the best angle for range is 45 degrees.

Formulas used

  • v_x = v_0\cos\theta,\quad v_y = v_0\sin\theta
  • H = h_0 + \frac{v_y^2}{2g}
  • R = v_x\,t_{flight}

Worked example

With the default numbers in the calculator above:

  1. Velocity components: 14.1 and 14.1 m/s
  2. Time of flight: 2.88 s
  3. Horizontal range: 40.8 m
  4. Maximum height above the landing level: 10.2 m

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What launch angle gives the longest range?

45 degrees when the launch and landing heights are equal. From a height, the best angle is a little lower.

Is air resistance included?

No. It reduces both range and height, especially for light or fast objects.

What value of g is used?

9.81 m/s².

Related calculators

Further reading: Meriam and Kraige, Engineering Mechanics: Dynamics; Hibbeler, Engineering Mechanics: Dynamics; Rao, Mechanical Vibrations.

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