Introduction

In the realm of calculus, the relationship between velocity and displacement is a fundamental concept. This article explores how to find the displacement of an object from t 1 second to t 4 seconds given its velocity vt -2t 4, where t is the time in seconds. By integrating the velocity function, we can determine the displacement of the object over the specified time interval.

Determining the Displacement Using Integration

The velocity of a moving object in meters per second is given by the function vt -2t 4. The displacement xt can be found by integrating the velocity function with respect to time:

t

xt int vtdt int (-2t 4) dt

This integral evaluates to:

xt -t2 4t C

where C is the constant of integration. To find the constant, we need to determine the initial conditions from which we can solve for C.

Evaluating the Displacement from t1s to t4s

First, we will evaluate the displacement at t 1 and t 4 seconds:

At t 1s:

x4 -42 4(4) C -16 16 C C

At t 1s, the displacement is:

x1 -12 4(1) C -1 4 C 3 C

To find the displacement from t 1s to t 4s, we subtract the displacement at t 1s from the displacement at t 4s:

x4 - x1 C - (3 C) -3

Therefore, the total displacement of the object from t 1s to t 4s is 3 units.

Further Explorations

For a more comprehensive understanding of calculus and its applications, consider the following:

tGraphical Interpretation: Visualize the velocity function as a graph and observe how the area under the curve between t 1s and t 4s corresponds to the displacement. tPhysical Interpretation: Interpret the mathematical result in the context of real-world motion, where the displacement represents the net change in position of the object. tError and Approximation: Discuss the types of errors that may arise when using numerical methods to approximate integrals. tRelated Problems: Solve similar problems where the velocity function is not constant but varies with time.

Conclusion

By integrating the velocity function, we have determined the displacement of the object from t1 second to t4 seconds. This process showcases the power of calculus in understanding the relationship between velocity and displacement. Practical applications in physics, engineering, and other fields depend on such mathematical tools to model and analyze motion.