Solve v = v₀ + at, visualize projectile arcs, and sketch v-t graphs to instantly see the corresponding x-t and a-t curves. From high school physics to CAE transient dynamics.
Mode
Enter 3 of 5 variables — solve for the rest
Initial position x₀ (m)
Initial velocity v₀ (m/s)
Final velocity v (m/s)blank = unknown
Acceleration a (m/s²)
Time t (s)blank = unknown
Result
Enter parameters above.
Presets
Equations of Motion
① $v = v_0 + at$
② $x = x_0 + v_0 t + \tfrac{1}{2}at^2$
③ $v^2 = v_0^2 + 2a(x - x_0)$
Launch conditions
Initial speed v₀ (m/s)30
Launch angle θ (°)45°
Gravity g (m/s²)9.81
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Range R (m)
—
Max height H (m)
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Flight time (s)
45°
Optimal angle
Projectile Formulas
Range: $R = \dfrac{v_0^2 \sin 2\theta}{g}$
Max height: $H = \dfrac{v_0^2 \sin^2\!\theta}{2g}$
Flight time: $T = \dfrac{2v_0 \sin\theta}{g}$
Draw a v-t Graph
Click or tap on the canvas to place points and build a piecewise v-t graph. The x-t (integral) and a-t (derivative) graphs are generated automatically.
Graph Relationships
Area under v-t → displacement x
Slope of v-t → acceleration a
Constant a: v-t is linear, x-t is parabolic.
CAE link: The Newmark-β method iteratively updates acceleration, velocity, and displacement at each time step — a direct generalization of these constant-acceleration equations to multi-DOF structural systems.
Position x(t)
Velocity v(t)
Acceleration a(t)
Trajectory, velocity vectors, and Vx/Vy components displayed simultaneously
Click to add points and draw your v-t graph
x-t graph (displacement)
a-t graph (acceleration)
v-t graph (velocity)
What is Kinematics?
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What exactly is kinematics? Is it just about plugging numbers into formulas?
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Basically, kinematics is the geometry of motion. It describes *how* things move—their position, velocity, and acceleration—without worrying about *why* they move (that's dynamics). In practice, the equations you see here are the mathematical toolkit for that description. Try moving the 'Initial Velocity' slider in the 1D section above and watch how the position-time graph changes instantly. That's kinematics in action.
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Wait, really? So if I know the acceleration and initial speed, these equations can tell me everything? What's the catch?
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The key catch is constant acceleration. These three famous equations only work when acceleration doesn't change. A common case is free fall near Earth's surface, where gravity provides a constant ~9.8 m/s² downward. For instance, if you throw a ball straight up, its acceleration is *always* -9.8 m/s², even at the top where its velocity is zero. Try setting acceleration to -9.8 in the simulator and see how velocity decreases linearly.
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That makes sense for 1D. But the simulator also does 2D projectiles. How do those equations connect? Is it just two 1D motions glued together?
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Exactly right! Projectile motion is the perfect example of superposition. The horizontal motion has zero acceleration (constant velocity), while the vertical motion has constant downward acceleration from gravity. They are independent. When you change the launch angle θ in the 2D panel, you're just splitting the initial velocity into horizontal ($v_0 \cos\theta$) and vertical ($v_0 \sin\theta$) components, then applying the 1D equations to each.
Physical Model & Key Equations
The core model assumes constant acceleration. The three equations form a complete set, allowing you to solve for any unknown kinematic variable if you know the others. They are derived from the definitions of velocity and acceleration.
Where $x_0$ is initial position (m), $v_0$ is initial velocity (m/s), $v$ is final velocity (m/s), $a$ is constant acceleration (m/s²), and $t$ is time elapsed (s).
For 2D projectile motion, we apply the constant-acceleration model separately to the x (horizontal) and y (vertical) axes. The only acceleration is gravity ($g$) acting downward. This leads to specific results for range and maximum height.
Here, $R$ is the total horizontal distance traveled (range), $H$ is the maximum height, $v_0$ is the initial launch speed, $\theta$ is the launch angle above the horizontal, and $g$ is the acceleration due to gravity (9.8 m/s² on Earth).
Real-World Applications
Automotive Safety & Crash Testing: CAE software uses the fundamental principles of kinematics as a starting point for simulating vehicle collisions. While real crashes involve complex, changing forces, the constant-acceleration equations provide a first-order estimate of stopping distances and the forces occupants experience, which informs airbag deployment timing and crumple zone design.
Sports Science & Performance: Analyzing the trajectory of a basketball, soccer ball, or javelin is pure projectile motion. By understanding the relationship between launch angle, speed, and range, coaches and athletes can optimize techniques for maximum distance or accuracy, such as finding the ideal angle for a long throw-in.
Robotics & Trajectory Planning: For a robotic arm to move smoothly from point A to point B, its controller must plan a kinematic trajectory—defining its position, velocity, and acceleration at every moment in time. These constant-acceleration equations form the basis for simple, smooth motion profiles.
Structural Dynamics & Earthquake Engineering: The CAE link mentioned—the Newmark-β method—is a direct generalization of these kinematics equations. It solves for the displacement, velocity, and acceleration of complex structures (like buildings or bridges) over time when subjected to dynamic loads like earthquakes, using the same core concepts of relating these three quantities.
Common Misconceptions and Points to Note
First, note that it's easy to forget the assumption of "constant acceleration". This simulator deals with an ideal world where "acceleration is constant". For example, even if you press a car's accelerator steadily, the actual acceleration changes due to air resistance or road gradient. In practice, you should treat calculation results from this tool as a "rough guideline" and verify them with more detailed models.
Next, calculation errors due to mixed units are frequent. The simulator's input fields are primarily in [m/s] or [m/s²], but real-world data often comes in [km/h] or [G]. For instance, to input an initial velocity of 60 km/h, you must convert it to 60 ÷ 3.6 ≈ 16.7 m/s. This kind of unit conversion mistake can throw your result off by an order of magnitude, so get into the habit of always checking your units.
Finally, there is a common misunderstanding of overlooking the "launch and landing point heights" in 2D projectile motion. The simulator's default setting assumes the launch and landing points are at the same height (y=0), but for example, when throwing a ball off a cliff, the initial height $y_0$ is positive. This difference significantly affects the time of flight and range. If you experiment with the altitude parameter in the tool, you should be able to appreciate the magnitude of this effect.
Related Engineering Fields
This constant acceleration motion calculation is the very foundation of trajectory planning in robotics. When designing a path for an industrial robot arm's end-effector to move smoothly from one point to another, you need to determine the velocity and acceleration profiles. The concept of the "v-t graph" you experimented with in NovaSolver is precisely the basics of robot motion speed planning (motion profiling).
It also forms the core of ADAS (Advanced Driver-Assistance Systems) development for automobiles. A simple model for estimating the Time To Collision (TTC) with a vehicle ahead calculates by assuming constant acceleration motion for the relative velocity and relative acceleration between your vehicle and the target. While more advanced CAE simulations add driver models and sensor noise, the core physics is what you learn here.
Going a step further, it connects to understanding response spectra in seismic design of structures. While seismic motion is complex, simplifying it, the foundation for calculating the inertial force (so-called seismic force) acting on a structure when the ground shakes with constant acceleration motion lies here. The constant acceleration motion formulas are the first door into the world of dynamic analysis.
For Further Learning
Once you're comfortable with NovaSolver, the next step is to consider the world where "acceleration is not constant". For example, in the motion of a mass attached to a spring (simple harmonic motion), acceleration changes proportionally with position $x$ ($a = -\omega^2 x$). To handle motion where acceleration is a function of velocity or position like this, knowledge of differential equations becomes essential. Understand that the high school physics "equation of motion $F=ma$" can be written more generally as $m\frac{d^2x}{dt^2}=F(x, v, t)$.
Regarding the mathematical background, your vision will broaden if you see the three constant acceleration motion formulas as a special case of the integral relationship from acceleration → velocity → position. Generally, when acceleration $a(t)$ is given, velocity $v$ is found by $v = v_0 + \int_0^t a(\tau) d\tau$, and position $x$ by $x = x_0 + \int_0^t v(\tau) d\tau$. In the case of constant acceleration ($a$=constant), executing these integrals yields exactly those three formulas. This perspective of "kinematics as integration" forms the foundation for understanding how CAE software solves complex motion through numerical integration (Euler's method, Runge-Kutta methods, etc.).
Recommended specific next topics are "projectile motion with air resistance" and "collision of two point masses (conservation of momentum)". The former lets you learn how the ideal parabola from NovaSolver becomes distorted in reality. The latter is the first step in broadening your perspective from the motion of a single object to the interaction of multiple objects. As you study these phenomena, be conscious that they too are ultimately described within the framework of the equations of motion; your knowledge should then connect into a coherent thread.