Relationship between Position and Velocity
Position Function
- Definition: The position function represents the location of an object at any given time. It is typically denoted by s(t) or x(t), where t represents time. For more on understanding the basics of position functions, see domain and range.
Velocity Function
- Definition: The velocity function represents the rate of change of position with respect to time. It is the derivative of the position function. Mathematically, v(t) = ds(t)/dt or v(t) = dx(t)/dt. Understanding the chain rule is essential for differentiating position functions.
- Implications:
- If the velocity is positive, the object is moving in the positive direction.
- If the velocity is negative, the object is moving in the negative direction.
- The magnitude of velocity gives the speed of the object, irrespective of its direction.
Applications in Motion Analysis
Determining Direction of Motion
By examining the sign of the velocity function, we can determine the direction of motion:
- Positive Velocity: If v(t) > 0, the object is moving in the positive direction.
- Negative Velocity: If v(t) < 0, the object is moving in the negative direction.
Example: Given the position function s(t) = t3 - 6t2 + 9t, find the intervals where the object is moving in the positive direction.
Solution:
- First, determine the velocity function: v(t) = 3t2 - 12t + 9.
- To ascertain where v(t) > 0, set 3t2 - 12t + 9 > 0 and solve for t. This will give the intervals where the object is moving in the positive direction. This method is similar to finding local extrema in functions.
Finding Resting Points
Resting points are crucial as they represent moments when the object momentarily stops before possibly changing its direction. At these points, the velocity is zero.
Example: Using the position function s(t) = t3 - 6t2 + 9t, determine the times when the object is at rest.
Solution:
- We already have the velocity function: v(t) = 3t2 - 12t + 9.
- Set v(t) = 0 to find the resting points. Solve for t to determine the times when the object is at rest. This solution can often be supported by proof by mathematical induction.
Determining Speed
Speed is the magnitude of velocity. It's always positive, even if an object is moving in the negative direction.
Example: For the position function s(t) = t3 - 6t2 + 9t, find the speed of the object at t = 2.
Solution:
- Using the velocity function v(t) = 3t2 - 12t + 9, evaluate v(2) to determine the velocity at t = 2.
- The speed is the absolute value of this velocity.
Real-World Applications
Astronomy
Astronomers frequently use the concepts of position and velocity to track the movement of celestial bodies, such as stars, planets, and comets. By understanding their velocities, astronomers can predict their future positions and study their past movements. This involves using techniques like definite integration to calculate distances travelled over time.
Sports
In sports, coaches and athletes use the principles of position and velocity to analyse and improve performance. For instance, in athletics, understanding an athlete's velocity can help in strategising for races. In team sports, analysing the motion of players can offer insights into game strategies.
Transport and Engineering
Engineers utilise the principles of position and velocity in designing transportation systems. Whether it's the motion of cars on roads, trains on tracks, or planes in the sky, understanding how objects move is crucial for safety and efficiency.
Environmental Studies
Scientists studying animal migration or the flow of rivers often use these concepts. By tracking the velocity of animals or water, they can make predictions about patterns, understand behaviours, and devise strategies for conservation.
FAQ
Yes, an object can have a zero velocity but a non-zero acceleration. A classic example is a ball thrown upwards. At the highest point of its trajectory, its velocity is zero because it momentarily stops before starting its descent. However, throughout its motion, including at the highest point, it experiences a downward acceleration due to gravity. This acceleration is what causes the ball to decelerate on its way up and accelerate on its way down. So, even when its velocity is zero at the peak, its acceleration remains non-zero.
Acceleration is the rate of change of velocity with respect to time. It tells us how quickly the velocity of an object is changing. If an object is speeding up, its acceleration is positive, and if it's slowing down, the acceleration is negative. Just as velocity is the derivative of the position function with respect to time, acceleration is the derivative of the velocity function with respect to time. In practical terms, when you press the accelerator pedal in a car, you increase its acceleration, leading to an increase in velocity. Conversely, when you apply brakes, you induce negative acceleration, causing the car to slow down.
Position-time and velocity-time graphs are visual tools that offer insights into the motion of an object. A position-time graph plots the position of an object against time. The slope of the graph at any point represents the velocity at that time. A steeper slope indicates a higher velocity. A velocity-time graph, on the other hand, plots the velocity of an object against time. The area under the curve of a velocity-time graph gives the displacement of the object. The slope of a velocity-time graph represents the acceleration. By analysing these graphs, one can gain a comprehensive understanding of an object's motion, including its speed, direction, and how these quantities change over time.
Instantaneous velocity refers to the velocity of an object at a specific instant in time. It is the rate of change of position at that exact moment. On the other hand, average velocity is the total displacement divided by the total time taken. In other words, it gives the average rate of change of position over a time interval. While instantaneous velocity can vary from moment to moment, average velocity gives a broader view of the motion over a period. For instance, a car might have varying speeds on a journey, but its average velocity will give a single value representing its overall performance on the trip.
Displacement refers to the overall change in position of an object over a certain time interval. It is the difference between the final and initial positions of the object. While position gives us the exact location of an object at a specific time, displacement gives us the 'net movement' from a starting point. Velocity, on the other hand, tells us the rate at which the position is changing. If you integrate the velocity function over a time interval, you get the displacement for that interval. In simpler terms, while velocity tells us how fast and in what direction an object is moving, displacement tells us how far and in what direction it has moved from its starting point.
Practice Questions
To find the velocity of the object at t = 3 seconds, we need to differentiate the position function with respect to t. The velocity function v(t) is given by the derivative of s(t). Differentiating s(t) with respect to t, we get: v(t) = 3t2 - 8t + 5. Evaluating this at t = 3, we get: v(3) = 3(32) - 8(3) + 5 = 27 - 24 + 5 = 8. Thus, the velocity of the object at t = 3 seconds is 8 m/s.
To determine when the car is stationary, we need to find when its velocity is zero. The velocity function v(t) is the derivative of the position function s(t). Differentiating s(t) with respect to t, we get: v(t) = 4t - 5. Setting this equal to zero to find when the car is stationary, we get: 4t - 5 = 0, 4t = 5, t = 1.25. Thus, the car is stationary at t = 1.25 seconds.
