Understanding displacement and velocity is fundamental to examining the motion of objects. These concepts describe how fast something is moving, its direction, and its overall change in position. They allow for accurate analysis and prediction of movement.
Understanding Displacement and Velocity
Displacement refers to an object’s overall change in position, representing the shortest straight-line distance from its starting point to its final destination. It includes both magnitude and direction, making it a vector quantity. For instance, if someone walks 5 meters east, their displacement is 5 meters east, regardless of the path taken. In contrast, distance is a scalar quantity that measures the total path covered by an object during its motion, without considering direction. A person walking 4 meters east, then 4 meters west, has a total distance traveled of 8 meters, but their displacement is zero because they returned to their starting point.
Similarly, velocity describes the rate at which an object changes its position, incorporating both speed and direction. It is a vector quantity, meaning a velocity of “5 meters per second east” specifies both magnitude and direction. Speed, on the other hand, is a scalar quantity that indicates only how fast an object is moving, without specifying its direction. An object moving at a constant speed in a circle has a continuously changing velocity because its direction is constantly shifting, even if its speed remains the same. Standard units for displacement are meters (m), while velocity is typically measured in meters per second (m/s).
Calculating Displacement with Constant Velocity
Determining displacement is straightforward when an object moves at a consistent velocity, maintaining a steady speed and unchanging direction. The fundamental relationship for calculating displacement under constant velocity is: Displacement = Velocity × Time, often written as d = v t.
For example, if a car travels at a constant velocity of 20 meters per second (m/s) east for 10 seconds, its displacement is 200 meters (20 m/s 10 s). This simple formula applies only when the velocity remains uniform throughout the entire duration of the motion.
Finding Displacement with Changing Velocity
When an object’s velocity is not constant, meaning it is either speeding up, slowing down, or changing direction, the situation involves acceleration. Acceleration is the rate at which an object’s velocity changes over time. In such cases, the simple d = v t formula is no longer sufficient to determine displacement. Instead, more comprehensive kinematic equations, which account for constant acceleration, become necessary.
One kinematic equation to find displacement when acceleration is constant is d = v₀t + (1/2)at², where ‘d’ is displacement, ‘v₀’ is initial velocity, ‘a’ is constant acceleration, and ‘t’ is time. For instance, if a bicycle starts from rest (v₀ = 0 m/s) and accelerates uniformly at 2 m/s² for 5 seconds, its displacement is 25 meters (0 m/s 5 s + 1/2 2 m/s² (5 s)²).
Displacement from Velocity-Time Graphs
Velocity-time graphs offer a visual and intuitive method for determining an object’s displacement. On such a graph, the area enclosed between the velocity curve and the time axis directly represents the object’s displacement over that specific time interval. This concept works because the units of the graph’s axes, velocity (meters per second) and time (seconds), multiply together to yield units of displacement (meters).
To calculate displacement from these graphs, find the area of the geometric shapes formed. For constant velocity, the area under the curve is a rectangle (velocity × time). If an object undergoes constant acceleration, the graph forms a triangle or a trapezoid, and their area formulas apply. For example, a triangular area with a base of 4 seconds and a height of 40 m/s represents a displacement of 80 meters (1/2 4 s 40 m/s).