Problem 21 - Applied Mathematics

Question

A particle is moving in a straight line. A displacement-time graph is drawn for its motion. The gradient of the tangent to the graph at time \(T\) represents the. velocity of the particle at time \(T\).

Step-by-Step Solution

Answer

The gradient of the tangent to the displacement-time graph at time T represents the velocity of the particle at time T.

Step 1: Identify the components involved

The problem involves a displacement-time graph of a particle moving in a straight line. We need to establish what the gradient of the tangent represents at a given time.

Step 2: Understand the displacement-time graph

A displacement-time graph shows how a particle's position changes over time. The horizontal axis represents time (T), and the vertical axis represents displacement.

Step 3: Define the gradient

The gradient of a graph at any point is the rate of change of the vertical axis variable with respect to the horizontal axis variable. In this graph, it represents the rate of change of displacement with respect to time.

Step 4: Relate the gradient to physical quantities

The rate of change of displacement with respect to time is the definition of velocity. Thus, the gradient of the tangent to the displacement-time graph at any time T gives the velocity of the particle at that time.

Key Concepts

velocity

Velocity represents how fast an object is moving and the direction in which it's moving. When dealing with a displacement-time graph, we are specifically looking at how the position of an object changes over time. In this context, velocity is a crucial concept. It tells us how quickly the displacement is changing as time progresses.

An important thing to remember about velocity is that it's a vector quantity. This means it has both a magnitude (speed) and a direction. In a straight line motion, this direction could simply be forward or backward. On a displacement-time graph, if the displacement is increasing with time, the velocity is positive. If the displacement is decreasing, the velocity is negative.

In essence, velocity helps us understand the motion of the particle by offering a snapshot of its movement at any given instant.

gradient of tangent

The gradient (or slope) of the tangent to a curve at any given point can tell us a lot about the behavior of the function at that point. In the case of a displacement-time graph, this gradient is particularly important.

To find the gradient, we draw a tangent line that just touches the curve at the point of interest. The steepness of this tangent line, measured as \(\frac{dy}{dx}\), tells us the rate at which displacement (y) is changing with respect to time (x).

Mathematically, if we have a function for displacement, say \( s(t) \), then the gradient of the tangent line at time \( T \) is given by \( \frac{ds}{dt} \). This gradient directly correlates to the concept of velocity - it tells us how fast and in what direction the displacement is changing at that specific moment.

rate of change

The rate of change is a fundamental concept in calculus and physics, which helps us understand how one quantity changes in relation to another. When examining a displacement-time graph, the rate of change of displacement with respect to time is essential.

This rate of change, represented by the gradient of the tangent to the curve, indicates how quickly the displacement is increasing or decreasing over time. In our context, this rate of change is none other than the velocity of the particle.

If the tangent at a certain point is steep, this means that the displacement is changing rapidly with time, resulting in a high velocity. If the tangent is flatter, the displacement is changing more slowly, leading to a lower velocity.

particle motion

The particle motion concept focuses on the dynamics of a particle as it moves along a path, which can be effectively analyzed using a displacement-time graph. By studying this graph, we can obtain a clear picture of how the particle's position changes with time.

This analysis is often broken down into understanding various aspects, such as:

• Displacement: The position of the particle at any given time.
• Velocity: The rate of change of displacement, found from the gradient of the tangent.
• Acceleration (if applicable): The rate of change of velocity, though this is beyond the basic displacement-time graph.

By focusing on these aspects, we can analyze the motion of any particle moving in a straight line. The displacement-time graph acts as a powerful tool, providing a visual and quantitative means to understand and describe this motion.