Problem 12-21: Engineering Mechanics- Dynamics (Hibbeler 14th Edition)

Kinematics with Exponential Velocity: Distance and Acceleration

This lesson applies integral and differential calculus to a velocity function of the form v(t) = 60(1 − e^(−t)). Students find the total distance traveled over a time interval by integrating velocity, then find instantaneous acceleration by differentiating velocity.

Question:
A freight train travels at a velocity v equals 60 times the quantity 1 minus e to the negative t, in feet per second, where t is the elapsed time in seconds. Determine the distance traveled in three seconds, and the acceleration at that time.

In this lesson, you will learn to:
• Compute the distance traveled over a given time interval by integrating a velocity function
• Derive the acceleration function by differentiating a given velocity function
• Evaluate definite integrals and derivatives involving natural exponential functions
• Interpret the ph