Why is there Acceleration due to a Change in the Direction of Velocity?
Acceleration is defined as the rate of change of velocity. The mathematical expression for acceleration is a dV/dT change in velocity/change in time.
The Nature of Vectors: Magnitude and Direction
Velocity and Acceleration are vectors, meaning they have both magnitude and direction. Any change in velocity in a particular direction leads to an acceleration in that direction. It is important to understand the distinction between speed and velocity: while speed is a scalar quantity that measures the rate of change in distance, velocity is a vector quantity that measures the rate of change in displacement.
Understanding Distance Traveled, Displacement, Speed, and Velocity
Distance Traveled: This is the total path length covered by a body from the point of origin to the point of destination. In the example given, a person traveling 4 meters eastward and then 3 meters northward covers a distance of 7 meters.
Displacement: This is the shortest path from the point of origin to the point of destination. In the same example, the person's displacement is 5 meters, 37 degrees north of east. The displacement indicates the direction and magnitude from the starting point to the endpoint.
Speed: This is the rate of change of distance covered. In the example, the person's speed is 0.7 m/s if they cover 7 meters in 10 seconds.
Velocity: This is the rate of change of displacement. If the person covered the displacement in 10 seconds, their velocity is 0.5 m/s.
Acceleration: This is the rate of change of velocity. It occurs when the direction of velocity changes even if the speed does not. In the example, the person's acceleration during the 180-degree turn is 0.05 m/s2.
Revisiting the Definition of Acceleration
The formula for acceleration is a dv/dt, where dv is the infinitesimal change in velocity and dt is the infinitesimal change in time during which the change in velocity occurred. While displacement does not directly appear in the formula for acceleration, it is used to determine both velocity and displacement.
In the equation v dx/dt, displacement x is used to determine velocity. Therefore, for velocity to exist, there must be a displacement. However, velocity can change from 0 to 0 or vice versa without any displacement, which can still generate acceleration. This is particularly evident in a 180-degree change in direction, where the velocity changes from a non-zero value to 0 and back again.
Conclusion
The key takeaway is that acceleration is fundamentally related to the change in velocity, regardless of whether displacement occurs. A change in direction is a prime example of this, where the velocity changes but no net displacement occurs.