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Tangential acceleration refers to the change in the speed of an object moving along a curved path. In this context, it's important to understand that it acts along the direction of motion itself, that is, tangential to the path. For the dog running counterclockwise on the circular path, its tangential acceleration is the rate at which its speed increases. Since the problem states the dog is speeding up at 5 feet per second squared, the tangential acceleration is a straightforward concept here—it is constant and equals 5 ft/s².

• This constant rate signifies how much faster the dog runs each second along the curve.
• Think of it like stepping harder on a gas pedal, causing the car to gain speed.

Understanding tangential acceleration is crucial, as it directly influences how rapidly the speed changes along any path the object follows.

Normal acceleration, often called centripetal acceleration, pertains to the change in direction of an object moving along a curved path. It acts perpendicular to the path, towards the center of the curvature, causing the object to turn or change its trajectory. For the circular path of radius 20 feet that the dog follows, the normal acceleration can be calculated using the formula:\[ a_N = \frac{v^2}{r} \]Where:

• \( v = 10 \text{ ft/s} \) is the speed of the object.
• \( r = 20 \text{ feet} \) is the radius of the circle.

Substituting the values, we find:\[ a_N = \frac{10^2}{20} = 5 \text{ ft/s}^2 \]This type of acceleration ensures the dog maintains a curved trajectory rather than moving in a straight line. It acts at all times along the radius inward—even when the dog does not change speed, guaranteeing its path remains circular.

Understanding vector components is essential for breaking down complex movements in two dimensions. In this scenario, the vector components of acceleration allow us to express the total acceleration in terms of the standard Cartesian unit vectors, \( \mathbf{i} \) and \( \mathbf{j} \). First, we establish the tangential (\( \mathbf{T} \)) and normal (\( \mathbf{N} \)) unit vectors, which characterize the movement along the curved path.

• The tangent vector, \( \mathbf{T} \), pointing in the direction of motion, is \( \left(-\frac{4}{5}, \frac{3}{5}\right) \).
• The normal vector, \( \mathbf{N} \), pointing inward, is \( \left(-\frac{3}{5}, -\frac{4}{5} \right) \).

Using these, any force or acceleration vector can be expressed in terms of \( \mathbf{T} \) and \( \mathbf{N} \), and subsequently in Cartesian coordinates by substituting:\[ \mathbf{a} = 5 (-\frac{4}{5} \mathbf{i} + \frac{3}{5}\mathbf{j}) + 5(-\frac{3}{5}\mathbf{i} - \frac{4}{5}\mathbf{j}) \]This breaks down to:\[ \mathbf{a} = -7\mathbf{i} - \mathbf{j} \]Understanding these components provides insight into how acceleration affects the dog in each direction on the plane, offering clarity on both the magnitude and direction of its overall acceleration.