use-integration-to-find-displacement
🚀 In kinematics, we study the motion of objects without considering the forces that cause the motion. One of the key aspects of kinematics is understanding how to find displacement, which is the change in position of an object. When the velocity of an object is given as a function of time, we can use calculus, specifically integration, to find the displacement over a given time interval. The displacement can be calculated by integrating the velocity function with respect to time.
Theory Explanation
Understanding Velocity and Displacement
Velocity is defined as the rate of change of displacement with respect to time. If we have a velocity function v(t), the displacement s over a time interval from t=a to t=b can be found by integrating the velocity function over that interval.
Key Points
- 🎯 Displacement is the integral of velocity over time.
- 🎯 The limits of integration correspond to the time interval for which displacement is calculated.
- 🎯 Understanding the relationship between velocity and displacement is crucial for solving kinematics problems.
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Examples:💡
Given the velocity function v(t) = 3t^2, find the displacement from t=1 to t=3.
Solution:
Step 1: Set up the integral for displacement: s = \int_{1}^{3} 3t^2 \, dt.
Step 2: Calculate the integral: s = [t^3]_{1}^{3} = 3^3 - 1^3 = 27 - 1 = 26.
Common Mistakes
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Mistake: Students often forget to apply the limits of integration correctly when calculating displacement.
Correction: Always ensure to substitute the upper and lower limits into the integrated function to find the correct displacement.
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Mistake: Confusing velocity with speed; students may forget that velocity is a vector quantity and can have negative values.
Correction: Remember that velocity includes direction, and negative values indicate motion in the opposite direction.