Momentum Calculator p = mv
Calculate momentum, impulse, collisions, and coefficient of restitution with step-by-step solutions. Perfect for GCSE and A-Level Physics.
Momentum = Mass × Velocity
Solve for:
Quick Examples
Momentum
kg·m/s (or N·s)
Impulse (force × time)
N·s (or kg·m/s)
Impulse-Momentum Theorem
N·s
Conservation of Momentum
kg·m/s
What is Momentum?
Momentum is the product of an object's mass and velocity. It describes how hard it is to stop a moving object. A heavy truck moving slowly can have the same momentum as a lightweight bullet moving very fast.
Momentum Formula
Momentum (kg·m/s) = Mass (kg) × Velocity (m/s)
Momentum is a vector quantity, meaning it has both magnitude and direction. If an object moves in the positive direction, its momentum is positive. If it moves in the negative direction, its momentum is negative.
Example: A 5 kg ball moving at 10 m/s
p = 5 × 10 = 50 kg·m/s
Real-World Momentum Examples
Walking person (70 kg, 1.5 m/s)
105 kg·m/s
Running sprinter (70 kg, 10 m/s)
700 kg·m/s
Bullet (0.01 kg, 400 m/s)
4 kg·m/s
Car at 30 mph (1500 kg, 13 m/s)
19,500 kg·m/s
Notice how the bullet and walking person have similar momentum despite vastly different masses and speeds!
Impulse and Change in Momentum
Impulse is the change in momentum of an object. It equals the force applied multiplied by the time the force acts. The impulse-momentum theorem connects force, time, and momentum change.
Impulse Formulas
J = FΔt = Δp = m(v - u)
Impulse (N·s) = Force (N) × Time (s) = Change in momentum
Key Facts
- • Impulse has units N·s or kg·m/s
- • Same units as momentum (they're equivalent!)
- • Area under a force-time graph = impulse
- • Impulse is a vector quantity
Why It Matters
- • Airbags: Increase time → reduce force
- • Crumple zones: Same principle
- • Bending knees: Softer landing
- • Cricket catch: Follow-through reduces force
💡 Exam Tip: If a question gives you a force-time graph, the impulse is the area under the curve. For constant force, area = F × Δt.
Conservation of Momentum
The Law of Conservation of Momentum states that in a closed system with no external forces, the total momentum before an interaction equals the total momentum after.
Conservation Equation
Total momentum before = Total momentum after
This law applies to ALL collisions, whether elastic or inelastic. It comes from Newton's Third Law: during a collision, the forces on each object are equal and opposite, so the momentum changes cancel out.
Why Momentum is Conserved
Newton's Third Law
F₁ on ₂ = -F₂ on ₁
Same contact time
Δt₁ = Δt₂
Equal & opposite impulses
Δp₁ = -Δp₂
Types of Collisions
Understanding the difference between elastic and inelastic collisions is crucial for A-Level Physics and beyond.
Elastic Collision
- ✓Momentum conserved
- ✓Kinetic energy conserved
- ✓Objects bounce apart
Examples: Newton's cradle, billiard balls (approximately), atomic collisions
Inelastic Collision
- ✓Momentum conserved
- ✗Kinetic energy NOT conserved
- ✗KE lost to heat, sound, deformation
Examples: Car crashes, catching a ball, most real-world collisions
Perfectly Inelastic (Objects Stick Together)
In a perfectly inelastic collision, the objects stick together and move as one unit after the collision. This is the collision type that loses the maximum kinetic energy.
Special formula (objects stick together):
m₁u₁ + m₂u₂ = (m₁ + m₂)v
Examples: Clay balls colliding, catching a ball, bullet embedding in a target, railway coupling
| Property | Elastic (e=1) | Inelastic (0<e<1) | Perfectly Inelastic (e=0) |
|---|---|---|---|
| Momentum conserved? | ✓ Yes | ✓ Yes | ✓ Yes |
| KE conserved? | ✓ Yes | ✗ No | ✗ No (max loss) |
| After collision | Bounce apart | May bounce | Stick together |
Coefficient of Restitution
A-LevelThe coefficient of restitution (e) measures how "bouncy" a collision is. It's the ratio of the relative speed of separation to the relative speed of approach.
Coefficient of Restitution Formula
e = (relative speed of separation) / (relative speed of approach)
Common e Values
| Collision Type | e Value | Examples |
|---|---|---|
| Perfectly elastic | e = 1 | Newton's cradle, ideal billiard balls |
| Elastic | e ≈ 0.9 | Tennis ball on hard court |
| Inelastic | e ≈ 0.5 | Football header, basketball |
| Inelastic | e ≈ 0.2 | Car crash (crumple zones) |
| Perfectly inelastic | e = 0 | Clay balls, bullet in wood |
📝 A-Level Note: The coefficient of restitution can also be used with the conservation of momentum to find final velocities: Use both equations simultaneously: (1) m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ and (2) e(u₁ - u₂) = v₂ - v₁
Common Mistakes in Momentum Problems
Avoid these frequent errors when solving momentum questions in GCSE and A-Level Physics exams:
Forgetting momentum is a vector
Momentum has direction! If one object moves right (+) and another moves left (-), you MUST use opposite signs. A 5 kg ball moving at 3 m/s LEFT has momentum -15 kg·m/s, not +15.
Always define a positive direction first. Objects moving in that direction get +v, opposite direction gets -v.
Confusing elastic and inelastic collisions
Students often think 'inelastic' means momentum isn't conserved. WRONG! Momentum is ALWAYS conserved in collisions. It's only kinetic energy that differs.
Momentum: conserved in ALL collisions. Kinetic energy: only conserved in ELASTIC collisions.
Thinking momentum = kinetic energy
Momentum (p = mv) and kinetic energy (KE = ½mv²) are different! Doubling velocity doubles momentum but QUADRUPLES kinetic energy. They have different units too.
Momentum is kg·m/s, KE is Joules. Momentum is a vector, KE is a scalar.
Using wrong signs for velocity direction
After a collision, objects might change direction. If object A was moving right (+3 m/s) and bounces back left, its final velocity is NEGATIVE (e.g., -2 m/s), not positive.
Draw a diagram! Show before and after, label directions, then assign +/- consistently.
Forgetting units for momentum
Momentum has units kg·m/s (same as N·s). Writing just '50' instead of '50 kg·m/s' loses marks in exams. Also, don't confuse with m/s (velocity units).
Always write units. kg·m/s or N·s are both acceptable for momentum.
Not checking if momentum is conserved
After solving, check your answer! Total momentum before should equal total momentum after. If not, you made a calculation or sign error.
Calculate total p before and p after. They must be equal (allowing for rounding).
Worked Examples
Practice with these GCSE and A-Level style momentum problems:
Example 1: Finding Momentum
A 5 kg ball moves at 10 m/s to the right. Calculate its momentum.
Solution:
Given: m = 5 kg, v = 10 m/s, p = ?
Formula: p = mv
p = 5 × 10
p = 50 kg·m/s (to the right)
Example 2: Average Force in a Collision
A 0.4 kg football is kicked and its velocity changes from 0 to 20 m/s. The contact time is 0.1 s. Find the average force.
Solution:
Impulse = change in momentum = m(v - u)
J = 0.4 × (20 - 0) = 8 N·s
Using J = FΔt, rearrange: F = J/Δt
F = 8 / 0.1
F = 80 N
Example 3: Perfectly Inelastic Collision
A 2 kg trolley moving at 3 m/s collides with a stationary 1 kg trolley. They stick together. Find the final velocity.
Solution:
Using conservation of momentum:
m₁u₁ + m₂u₂ = (m₁ + m₂)v
(2 × 3) + (1 × 0) = (2 + 1) × v
6 = 3v
v = 2 m/s
Example 4: Elastic Collision with Equal Masses
In an elastic collision, a 2 kg ball moving at 4 m/s hits a stationary 2 kg ball. Find the final velocities.
Solution:
For equal masses in an elastic collision:
The objects "swap" velocities!
Ball 1: v₁ = 0 m/s (stops)
Ball 2: v₂ = 4 m/s (takes all the momentum)
This is why Newton's cradle works - each ball transfers its momentum to the next.
Example 5: Finding the Coefficient of Restitution
Ball A (u₁ = 5 m/s) collides with ball B (u₂ = 0 m/s). After collision: v₁ = 1 m/s, v₂ = 4 m/s. Find e.
Solution:
e = (v₂ - v₁) / (u₁ - u₂)
e = (4 - 1) / (5 - 0)
e = 3 / 5
e = 0.6
Since 0 < e < 1, this is an inelastic collision (some KE lost).
Frequently Asked Questions
What is momentum?
Momentum is the product of mass and velocity (p = mv). It describes how hard it is to stop a moving object. It is a vector quantity measured in kg·m/s.
What is the formula for momentum?
p = mv, where p is momentum (kg·m/s), m is mass (kg), and v is velocity (m/s). Since velocity is a vector, momentum is also a vector.
What are the units of momentum?
Momentum is measured in kg·m/s (kilogram-metres per second), which is equivalent to N·s (Newton-seconds). Both units are acceptable.
What is impulse in physics?
Impulse (J) is the change in momentum. It equals force × time (J = FΔt) or mass × change in velocity (J = mΔv). Units are N·s or kg·m/s.
What is conservation of momentum?
In a closed system, total momentum before = total momentum after. This applies to ALL collisions: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
Elastic vs inelastic collisions?
Elastic: both momentum AND kinetic energy conserved (objects bounce). Inelastic: only momentum conserved, some KE lost to heat/sound/deformation.
What is coefficient of restitution?
e measures how bouncy a collision is: e = (v₂-v₁)/(u₁-u₂). e=1 is perfectly elastic, e=0 is perfectly inelastic (objects stick).
Is this calculator suitable for A-Level?
Yes! It covers momentum, impulse, elastic/inelastic collisions, and coefficient of restitution with step-by-step solutions for A-Level Physics.
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