Impulse and Momentum
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Core Definitions & Impulse-Momentum Theorem
Linear Momentum (p)
p = mv
A vector quantity. Units: kg m/s. Direction is same as velocity.
Impulse (J)
J = Fₐᵥ ⋅ Δt = ∫ F dt
The product of average force and time duration. It's a vector.
Impulse-Momentum Theorem
J = Δp = p_f - pᵢ
The impulse on an object equals its change in momentum.
Newton's 2nd Law (Momentum Form)
Fₑₓₜ = dp/dt
The net external force is the rate of change of momentum.
Graphical Impulse
J = Area under F-t graph
Calculate the area of the force vs. time plot to find total impulse.
Conservation of Momentum & Applications
Principle of Conservation
If Fₑₓₜ = 0, then Δp_system = 0
p_initial = p_final
p_initial = p_final
For an isolated system, the total linear momentum remains constant.
System of Particles
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
The vector sum of momenta before interaction equals the vector sum after.
Recoil of a Gun
m_g v_g = -m_b v_b
Gun and bullet have equal and opposite momenta. System momentum remains zero.
Explosion of a Bomb
p₁ + p₂ + p₃ + ... = 0
If initially at rest, the vector sum of the momenta of all fragments is zero.
Collisions & Coefficient of Restitution
Coefficient of Restitution (e)
e = |v₂ - v₁| / |u₁ - u₂|
e = (Rel. speed of separation) / (Rel. speed of approach)
e = (Rel. speed of separation) / (Rel. speed of approach)
A dimensionless number that characterizes the elasticity of a collision.
Velocities after 1D Elastic Collision (e=1)
v₁ = (m₁-m₂)u₁/(m₁+m₂) + 2m₂u₂/(m₁+m₂)
v₂ = 2m₁u₁/(m₁+m₂) + (m₂-m₁)u₂/(m₁+m₂)
v₂ = 2m₁u₁/(m₁+m₂) + (m₂-m₁)u₂/(m₁+m₂)
Memorize the special cases, not these long formulas.
Loss in Kinetic Energy
ΔKE_loss = ½ (m₁m₂/(m₁+m₂)) (u₁-u₂)² (1-e²)
Max loss occurs when e=0 (perfectly inelastic collision).
BITSAT Collision Speed Hacks
For elastic collisions (e=1) with equal masses (m₁=m₂), velocities are simply exchanged: v₁=u₂ and v₂=u₁.
If a heavy body hits a light body at rest (m₁ >> m₂, u₂=0), the heavy body continues almost unaffected (v₁≈u₁) and the light body moves at twice the speed (v₂≈2u₁).
If a light body hits a heavy body at rest (m₁ << m₂, u₂=0), the light body rebounds with the same speed (v₁≈-u₁) and the heavy body barely moves (v₂≈0).
In perfectly inelastic collisions (e=0), bodies stick together. Use momentum conservation: m₁u₁ + m₂u₂ = (m₁+m₂)v_final.
Center of Mass (CM)
CM for Discrete Particles
x_cm = (Σmᵢxᵢ) / (Σmᵢ)
y_cm = (Σmᵢyᵢ) / (Σmᵢ)
y_cm = (Σmᵢyᵢ) / (Σmᵢ)
Weighted average of positions. For a two-particle system: x_cm = (m₁x₁ + m₂x₂) / (m₁ + m₂).
Velocity of CM
V_cm = (m₁v₁ + m₂v₂ + ...) / (m₁ + m₂ + ...)
The total momentum of a system is the total mass times the velocity of the CM: p_sys = M_total * V_cm.
Motion of CM
M_total ⋅ A_cm = F_ext_net
The CM moves as if all the system's mass were concentrated at that point and all external forces acted on it.
CM of Common Shapes
Semicircular ring: 2R/π
Semicircular disc: 4R/3π
Hemisphere: 3R/8
Solid Cone: h/4
Semicircular disc: 4R/3π
Hemisphere: 3R/8
Solid Cone: h/4
Positions are from the base. These are frequently asked.
Collision Types at a Glance
| Property | Elastic Collision | Inelastic Collision | Perfectly Inelastic Collision |
|---|---|---|---|
Coefficient (e) | e = 1 | 0 < e < 1 | e = 0 |
Momentum (p) | Conserved | Conserved | Conserved |
Kinetic Energy (KE) | Conserved | Not Conserved (Lost) | Not Conserved (Max Loss) |
Bodies after impact | Separate | Separate | Stick Together |