Solve the problem.
Given the velocity and initial position of a body moving along a coordinate line at time , find the
body’s position at time .

To find the body’s position at time , we start with the given velocity function . The position can be found by integrating the velocity function:

Using the substitution , we have , or . The integral becomes:

Now we need to determine the constant using the initial condition :

Since :

Solving for :

Thus, the position function is:

Among the options given, the correct position function is

Now, let’s keep your mind buzzing with some fun tidbits!

The concept of position, velocity, and acceleration dates back to the early thinkers in physics like Galileo and Newton. They laid the groundwork for the kinematics we use today. Imagine their excitement as they untangled how objects move, all without the benefit of our modern technology!

When approaching problems involving motion, a common mistake is skipping the integration or miscalculating the constants of integration. Always remember to double-check your initial conditions! It’s like baking a cake—every ingredient counts!