Finding Acceleration Magnitude at a Point

Full transcript

We are given the velocity of a particle as a vector function of x, y, and z, and we need to find the magnitude of acceleration at the specific point (1, 2, 4). The acceleration vector is the total derivative of velocity with respect to time, which expands using the chain rule into partial derivatives multiplied by the respective velocity components. We extract each component of velocity: the x-component, the y-component, and the z-component, and then compute each corresponding component of the acceleration vector separately. After applying the chain rule carefully to each component, we assemble the full acceleration vector, which contains terms involving the coordinates x, y, and z of the particle's position. Now we substitute the given point coordinates x equals 1, y equals 2, and z equals 4 into each component of the acceleration vector to get the numerical values at that specific location. Finally, we calculate the magnitude by taking the square root of the sum of the squares of all three acceleration components, giving us the final answer of 5 meters per second squared.