Arizona State University| PHY 121: Univ Physics I: Mechanics|Homework 1-1| Problem 4| Finding the Acceleration Vector

Learning Goal: To practice finding the acceleration vector

Suppose an object has an initial velocity \vec{v_{i}} at time t_i and later, at time t_f, has velocity \vec{v_{f}}. The fact that the velocity changes tells us the object undergoes an acceleration during the time interval \Delta \:t=t_f-t_i. From the definition of average acceleration, 

\vec{a}=\frac{\vec{v_f}-\vec{v_i}}{t_f-t_i}=\frac{\Delta \vec{v}}{\Delta t},

we see that the acceleration vector points in the same direction as the vector \Delta \vec{v}. This vector is the change in the velocity \Delta \vec{v}=\vec{v_f}-\vec{v_i}, so to know which way the acceleration vector points, we have to perform the vector subtraction \vec{v_f}-\vec{v_i}. This problem shows how to use vector subtraction to find the acceleration vector.

Three dots in a straight line are shown. From the left the first two dots are joined by a vector arrow labelled vector v sub i. A longer vector v sub f joins the second dot to the third.

Finding the acceleration Vector

To find the acceleration as the velocity changes from \vec{v_i} to \vec{v_f}:

    1. Draw the velocity vector \vec{v_f}

2. Draw \vec{-v_i} at the tip of \vec{v_f}

The vector v sub f from Figure 1 is shown starting at a dot. The vector minus v sub i is shown starting at the tip of vector v sub f parallel to it and pointing back along it.

3. Draw \Delta \vec{v}=\vec{v_f}-\vec{v_i}=\vec{v_f}+\left(-\vec{v_i}\right). This is the direction of \vec{a}

The vector v sub f from Figure 1 is shown starting at a dot. The vector minus v sub i is shown starting at the tip of vector v sub f parallel to it and pointing back along it. A third vector, Delta v, is shown drawn parallel to the others starting at the tail of vector v sub f and ending at the tip of vector v sub i.

4. Return to the original motion diagram. Draw a vector at the middle point in the direction of \Delta \vec{v}; label it \:\vec{a}. This is the average acceleration at the midpoint between \vec{v_i} and \vec{v_f}

The vectors in Figure 1 are shown with the acceleration vector pointing to the right parallel to the velocity vectors.

 

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