Lecture on Impulse Momentum Theorem

Overview

• Discussion on the Impulse Momentum Theorem
• Focus on a time-dependent force applied to a mass
• Objective: Find the speed of the object after 4 seconds

Key Concepts

Impulse Momentum Theorem

• Impulse: Equal to the change in momentum
• Formula: Impulse = Force × Time
• Integration over the entire interval gives the impulse

Time-Dependent Force

• Force is not constant; it varies with time
• Applied to a mass starting from rest

Problem Statement

• Mass starts at rest (initial velocity = 0)
• Force applied is time-dependent
• Need to find the speed after 4 seconds using impulse momentum theorem

Solution Approach

Impulse Calculation

• Impulse = Change in Momentum
• Change in momentum = Final momentum - Initial momentum

Integration

• Integrate the force over time to find the impulse
• Use the formula: If ( F(t) = t^n ), then ( \int F(t) dt = \frac{t^{n+1}}{n+1} )
• Note: The exponent increases by 1 during integration

Specific Integration

• Constants: 20 becomes ( \frac{t^3}{3} ), 50 becomes ( \frac{t^2}{2} )
• Time interval from 0 to 4 seconds

Calculations

• Set up equation: ( 2 \times V_f = 20/3 \times 4^3 + 25 \times 4^2 )
• Solve for ( V_f )
• Result: Final velocity ( V_f = 41.3 \text{ m/s} )

Conclusion

• The speed of the particle after 4 seconds is 41.3 m/s
• Used the impulse momentum theorem to derive the solution

Closing Remarks

• Importance of understanding time-dependent forces
• Reminder to like, share, and subscribe