45
N·s
225
N
15
kg·m/s
60
kg·m/s
45
kg·m/s
9
m/s
45
m/s²
337.5
J
337.5
J
45
N·s
225
N
15
kg·m/s
60
kg·m/s
45
kg·m/s
9
m/s
45
m/s²
337.5
J
337.5
J
The Impulse-Momentum Calculator bridges the two sides of the impulse-momentum theorem by computing both the impulse and the average force from velocity changes. Given an object's mass, initial velocity, final velocity, and the time interval of interaction, it calculates everything you need:
$$J = m(v_2 - v_1) = \Delta p$$
$$F_{\text{avg}} = \frac{\Delta p}{\Delta t} = \frac{m(v_2 - v_1)}{\Delta t}$$
This is the complete picture of how forces change motion over time. Whether you are analyzing a baseball bat hitting a ball, a car crash, a rocket exhaust, or a tennis serve, this calculator reveals the impulse delivered, the average force involved, and the energy change. The average force is especially useful because real-world forces vary during a collision — knowing the average gives you a practical single number for engineering and safety calculations.
The calculator also shows initial and final momenta separately, plus the change in kinetic energy, allowing you to see both the momentum and energy perspectives of the same event.
Starting from the impulse-momentum theorem:
Impulse: $$J = m(v_2 - v_1) = m \Delta v$$
Average Force: $$F_{\text{avg}} = \frac{J}{\Delta t} = \frac{m(v_2 - v_1)}{\Delta t}$$
Initial Momentum: $$p_i = mv_1$$
Final Momentum: $$p_f = mv_2$$
Kinetic Energy Change: $$\Delta KE = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$$
Note that $$\Delta KE$$ is not equal to $$F_{\text{avg}} \cdot \text{distance}$$ unless the force is truly constant. However, $$J = F_{\text{avg}} \cdot \Delta t$$ is exact by definition of the average.
A positive impulse means the object gained forward momentum (sped up or reversed from backward to forward). A negative impulse means it lost forward momentum (slowed down or reversed). The average force tells you the equivalent constant force that would produce the same momentum change — real forces during collisions can peak at many times this average. The kinetic energy change shows whether the interaction added or removed energy from the object.
Inputs
Results
A 5 kg object accelerated from 3 to 12 m/s in 0.2 s receives 45 N·s of impulse from an average force of 225 N.
Inputs
Results
A 1500 kg car braking from 20 m/s to rest over 4 s requires an average braking force of 7500 N and loses 300 kJ of kinetic energy.
The impulse-momentum theorem states $$J = \Delta p = m(v_f - v_i)$$. It connects the net impulse (force × time) to the resulting change in momentum. This theorem is derived directly from Newton's second law and holds for any force profile, not just constant forces.
Divide the impulse by the time interval: $$F_{\text{avg}} = J / \Delta t = m(v_2 - v_1) / \Delta t$$. This gives the constant force that would produce the same impulse. The actual instantaneous force may vary greatly during the interaction.
Real collision forces change rapidly and peak values are hard to measure directly. The average force gives a single, practical number for engineering calculations. It is used in crash safety analysis, sports biomechanics, and structural impact assessment.
Only if the velocity does not change ($$v_1 = v_2$$), meaning the object returns to its original state. In practice, this is rare for a net force. However, if multiple forces act and their impulses cancel, the net impulse is zero.
In an isolated system (no external forces), the total impulse is zero, so total momentum is conserved. The impulse-momentum theorem applies to individual objects within the system — each object's momentum changes by the impulse it receives, but the total remains constant.
Momentum and kinetic energy are different quantities. In an inelastic collision, total momentum is conserved but kinetic energy is converted to heat, sound, or deformation. Only in perfectly elastic collisions are both conserved simultaneously.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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