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Initial Setup
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- Consider an object of mass m moving in a straight line. It has an initial velocity u and gets uniformly accelerated to velocity v over time t due to a constant force F.
- Initial momentum: p1 = mu
- Final momentum: p2 = mv
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Change in Momentum
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The change in momentum (Δp) ∝ p2 – p1
Δp ∝ mv−mu
Δp ∝ m × (v − u)
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Rate of Change in Momentum
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- The rate of change is momentum
(Δp / t) ∝ m × (v − u) / t
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Applied Force
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F ∝ m × (v − u) / t
F = km × (v − u) / t
F = kma
Here, k is the constant of proportionality
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SI Units
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- Unit of mass is kilogram (kg) and unit of acceleration is m/s2
- One unit of force is defined such that k becomes one. Hence, 1 unit of force = k × (1kg) × (1ms -2).
- Therefore, F = ma (which is the mathematical representation of the second laws of motion). Newton (N) is the unit of Force.
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Real-life Observations:
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- Catching in Cricket: A cricket fielder pulls their hands back while catching a ball to increase the time taken for the ball’s velocity to reduce, decreasing the ball’s acceleration and impact.
- High Jump Landing: In a high jump event, athletes fall on cushioned or sand beds to increase the time taken for their momentum to stop, decreasing the change in momentum and the force of impact.
- Karate Ice Break: A karate player’s ability to break a slab of ice in one blow can also be explained using the second laws of motion.
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Relation with First Laws of Motion:
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- The first law (i.e. objects remain in their state of motion or rest unless acted upon by an external force) can be derived from the second law’s mathematical expression.
- Given F = ma, if F = 0, then v = u for any time t. If the object starts at rest (u = 0), it will remain at rest (v = 0).
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