动量计算器
This momentum calculator uses the formula p = m × v to solve for momentum (kg·m/s), mass (kg), or velocity (m/s). It also includes a collision calculator that applies conservation of momentum to find the final velocity after a perfectly inelastic collision. Choose a mode, enter two known values, and get an instant result.
Momentum Calculator
Use the momentum formula p = m × v to solve for momentum, mass, or velocity. Or switch to the Collision mode to find the final velocity after a perfectly inelastic collision.
Frequently Asked Questions
如何计算物体的动量？
动量 p = m × v，其中 m 是物体质量（kg），v 是速度（m/s），动量的 SI 单位为 kg·m/s（即 N·s）。动量是矢量，方向与速度方向相同。
动量守恒定律是什么？
在没有外力（或外力合力为零）的封闭系统中，系统总动量保持不变。公式为：m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'。该定律适用于碰撞分析：完全弹性碰撞（动量和动能均守恒）、完全非弹性碰撞（动量守恒，动能不守恒）。
冲量和动量变化有什么关系？
冲量 J = F × Δt（力乘以作用时间），等于动量的变化量 Δp = m × Δv。即冲量-动量定理：J = Δp。这解释了为什么安全气囊通过延长碰撞时间来降低平均作用力，从而减少伤害。
动量在碰撞分析中有什么应用？
动量守恒是分析碰撞的核心工具。在交通事故重建中，可通过车辆碰撞后的速度和方向反推碰撞前的速度；在台球、航天器对接和粒子物理实验中，动量守恒定律都有重要应用。
Understanding Momentum
The p = mv Formula
Momentum (p) is a measure of the quantity of motion an object has. It depends on two things: how heavy the object is and how fast it is moving. The standard SI formula is:
Momentum Formula:
p = m × v
- p = Momentum (kilogram metres per second, kg·m/s)
- m = Mass (kilograms, kg)
- v = Velocity (metres per second, m/s)
Unlike kinetic energy, momentum is a vector quantity — it has both magnitude and direction. A ball moving to the right has positive momentum; the same ball moving to the left has negative momentum (if we define rightward as positive). This sign convention is essential when analysing collisions.
The formula can be rearranged to solve for any of its three variables:
Solve for mass:
m = p / v
Solve for velocity:
v = p / m
Momentum is directly proportional to both mass and velocity — doubling either one doubles the momentum. This linear relationship distinguishes it from kinetic energy, which grows quadratically with velocity.
Conservation of Momentum
One of the most powerful principles in physics is the Law of Conservation of Momentum: in a closed system with no external forces, the total momentum before an event equals the total momentum after.
p_before = p_after
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
where v₁' and v₂' are the post-collision velocities.
This law applies in all collisions — elastic, inelastic, and perfectly inelastic. It also governs explosions, rocket propulsion, and recoil. For example, when a gun fires a bullet, the gun recoils with equal and opposite momentum to the bullet, keeping total momentum at zero (assuming the system started at rest).
Conservation of momentum is a consequence of Newton's third law: for every action there is an equal and opposite reaction. The forces exerted between colliding objects are equal in magnitude and opposite in direction, so they cancel out over the collision duration, leaving total momentum unchanged.
Elastic vs Inelastic Collisions
All collisions conserve momentum, but they differ in whether kinetic energy is conserved:
Elastic Collision
Both momentum and kinetic energy are conserved. Objects bounce off each other without permanent deformation. Examples: collisions between gas molecules, billiard balls (approximately). In practice, perfectly elastic collisions rarely occur at the macroscopic level.
Inelastic Collision
Momentum is conserved but kinetic energy is not. Some energy is converted to heat, sound, or deformation. Most real-world collisions fall into this category — car crashes, sporting impacts, everyday bumps and drops.
Perfectly Inelastic Collision
The objects stick together after impact and move as one combined mass. The maximum possible kinetic energy is lost while momentum is still conserved. The final velocity is:
v_f = (m₁v₁ + m₂v₂) / (m₁ + m₂)
Examples: clay balls colliding, coupling railway carriages, a bullet embedding in a block.
Real-World Examples
Football Tackle
A 90 kg defender running at 6 m/s tackles a 75 kg ball carrier at rest. What is the combined velocity after a perfectly inelastic collision?
p_before = 90 × 6 + 75 × 0 = 540 kg·m/s
v_f = 540 / (90 + 75) = 540 / 165 ≈ 3.27 m/s
About 3.27 m/s — the combined players continue moving in the tackle direction, demonstrating how momentum is shared in proportion to total mass.
Recoil of a Rifle
A 4 kg rifle fires a 10 g (0.01 kg) bullet at 900 m/s. What is the recoil velocity of the rifle? (System starts at rest: p_total = 0.)
0 = 0.01 × 900 + 4 × v_rifle
v_rifle = −9 / 4 = −2.25 m/s
The rifle recoils at 2.25 m/s in the opposite direction. The negative sign indicates direction — momentum is perfectly conserved.
Car Crash (Head-On)
A 1,200 kg car moving at 20 m/s collides head-on with a 1,000 kg car at 15 m/s (in the opposite direction, so v₂ = −15 m/s).
p_total = 1200 × 20 + 1000 × (−15) = 24,000 − 15,000 = 9,000 kg·m/s
v_f = 9,000 / (1200 + 1000) = 9,000 / 2200 ≈ 4.09 m/s
The wreckage moves at about 4.09 m/s in the direction of the heavier car. A large kinetic energy reduction confirms this is a highly inelastic event.
Rocket Propulsion
A 500 kg rocket at rest ejects 10 kg of exhaust at 2,000 m/s. By conservation of momentum:
0 = 10 × (−2000) + 490 × v_rocket
v_rocket = 20,000 / 490 ≈ 40.8 m/s
The rocket accelerates to about 40.8 m/s. Rocket engines apply this principle continuously — ejecting mass at high speed to gain momentum in the opposite direction.