Acceleration Acceleration measures the change in velocity (which is a vector) over time. This will be another vector. The derivative of velocity is acceleration or the second derivative of position. Curvature Curvature measures the amount of “bend” a curve has. This is done by measuring the total change in the direction (and only the direction) for each unit of length you move around the curve. Computing it this way is difficult though. Using the chain rule we can take the derivative of the unit tangent vector with respect to time. Taking the length of this vector and keeping in mind speed is a scalar we get the identity to the right. Keeping in mind the unit tangent vector T is the velocity divided by the speed multiplying the unit tangent by speed given velocity.
Acceleration, Speed, Curvature, Unit Tangent and Unit Normal Vectors The values of acceleration, speed, curvature, unit tangent and unit normal vectors can be related by a single equation. This can be done by taking the derivative of velocity and apply the product rule. From this we get the fundamental motion equation that gives the amount of forced directed in both the tangential and normal direction.
The motion equation can be used to derive a formula for curvature by crossing each side with velocity. Distribute the velocity. Velocity is parallel to the unit tangent vector so the cross product is zero. Solve this equation to get the curvature is the length of velocity crossed with acceleration divided by speed cubed.
Cross the velocity and acceleration and compute the length of this vector.
y x z In each case notice that the curvature is constant (a circle is “bent” the same way no matter where you look at it) and the value is the reciprocal of the radius of the circle.
Geometric Interpretation of Curvature The curvature at a point on a curve is the reciprocal of the circle of “best fit” to the curve at that point. If the curve is nearly straight the circle will need to be big so the reciprocal of the radius is small and if the curve has a lot of bend the circle is small so the reciprocal of the radius is big. small curvature large curvature
To find the place where the curvature is greatest (i.e. the turn is the tightest) in this situation is where the derivative is zero. If the car does not slide here it will not slide anywhere on this curve. Compute the curvature at this position. Recall the normal component of acceleration is given to the right.