 # Second Law of Motion: Momentum and Impulse

• According to second law of motion the rate of change of linear momentum is directly proportional to the external force applied on it.
• Newton’s second law of motion gives the formula of force f = ma.
• Second law implies that when a bigger force is applied on a body its linear momentum changes fast.
• Linear momentum is defined as the product of mass and velocity P = mv.
• Momentum in a vector quantity.
• Relation between F and P is \tt F \propto \frac{dp}{dt}
• 1 Newton = 105 dynes.
• \tt F = \frac{m(v - u)}{t} where, v = final velocity, u = inertial velocity.
• If “v” is constant and ‘m’ is changing \tt F = V\frac{dm}{dt}
• \tt F = \frac{P_{2} - P_{1}}{t} where, P1 = initial momentum, P2 = final momentum
• When a large force is acting on a small internal of time the product of force and time is called Impulse (J).
• The impulse J = f(t) = ∫fdt is a quantity that combines the net force and the time interval over which the force acts.
• Area under force time graph gives impulse.
• If a gun fires n bullets of mass ‘m’ \tt F = \frac{nmv}{t}
• If water of density ‘ρ’ coming out of pipe of area of cross section A with speed ‘v’     F = Aρv2
• The reaction force on a person in a lift moving up with acceleration R = m (g + a)
• The reaction force on a person in a lift moving down with deceleration R = m (g + a)
• The reaction force on a person in a lift moving up deceleration R = m (g − a)
• The reaction force on a person in a lift moving down acceleration R = m (g − a)
• The reaction force on a person in a freely falling lift R = 0.
• The force on between two bodies at contact is called contact force.
• Contact force \tt F = \frac{M_{2} F}{M_{1} + M_{2}} • Acceleration of two bodies system \tt a = \frac{F}{M_{1} + M_{2}}
• For three bodies \tt a = \frac{F}{m_{1} + m_{2} + m_{3}} • Tension is an electromagnetic force in a string due to force.
• Acceleration of two bodies connected by string \tt a = \frac{F}{m_{1} + m_{2}}
• Tension in string \tt T = \frac{m_{2} F}{m_{1} + m_{2}}
• Acceleration of three bodies connected by string \tt a = \frac{F}{m_{1} + m_{2} + m_{3}}
• Tension in first string \tt T_{1} = \frac{m_{1} F}{m_{1} + m_{2} + m_{3}}
• Tension in second string \tt T_{2} = \frac{\left(m_{1} + m_{2}\right) F}{m_{1} + m_{2} + m_{3}}
• At woods machine acceleration \tt a = \frac{m_{1} - m_{2}}{m_{1} + m_{2}} g • Tension \tt T = \frac{2 m_{1}m_{2}}{m_{1} + m_{2}} \cdot g
• Thrust on the pulley = \tt 2T = \frac{4 m_{1}m_{2} g}{m_{1} + m_{2}}
• Acceleration \tt a = \frac{m_{1} g}{m_{1} + m_{2}} •  Tension in string \tt T = \frac{m_{1} m_{2} g}{m_{1} + m_{2}}
• Acceleration \tt a = \frac{\left(m_{1} - m_{2} \sin \theta \right) g}{m_{1} + m_{2}} •  Tension \tt T = \frac{m_{1} m_{2} \left(1 + \sin \theta \right)}{m_{1} + m_{2}}
• Acceleration \tt a = \frac{\left(m_{1} \sin \alpha - m_{2} \sin \beta \right) g}{m_{1} + m_{2}} •  Tension \tt T = \frac{m_{1} m_{2} \left(\sin \alpha + \sin \beta \right) g}{m_{1} + m_{2}}

### View the Topic in this video From 0:36 To 44:05

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1. Newton's Second Law of Motion: \tt F = \frac{mdv}{dt} = ma

2. Impulse = Force × Time = Change in momentum

3. Position dependent force: Gravitational force between two bodies \frac{Gm_{1}m_{2}}{r^{2}}