Question

Triple Choice The position-time graph for the motion of a certain particle is a smooth curve, like a parabola. At a given instant of time, the tangent line to the positiontime graph has a negative slope. Is the instantaneous velocity of the particle at this time positive, negative, or zero? Explain.

Short answer

The instantaneous velocity is negative.

Step-by-step solution

Understanding the Graph

The position-time graph is described as a smooth curve like a parabola. A parabolic graph indicates that the particle is accelerating or decelerating over time. According to the problem, the tangent to this curve at a given instant has a negative slope.

Slope of Tangent and Velocity Relationship

Know that the slope of the tangent line to a position-time graph at any point gives the instantaneous velocity of the particle at that instant. A negative slope of the tangent line implies a negative value for the velocity.

Conclusion from the Slope

Since the slope of the tangent line is negative, and since this slope represents the instantaneous velocity, the instantaneous velocity of the particle is negative at the given instant in time.

Key concepts

Understanding Position-Time Graphs

A position-time graph is a powerful tool that provides insights into the movement of an object. It plots the position of the particle on the y-axis and the time on the x-axis. With this graph, you can determine how fast something is moving and whether it's speeding up or slowing down.

Typically, a straight line indicates uniform motion, where the particle moves at a constant speed. However, if the graph is a curve, such as a parabola, it suggests that the particle is undergoing acceleration or deceleration. In these situations, the curve's shape and slope become incredibly informative about the particle’s velocity at any given moment.

In the current exercise, the graph resembles a parabola, which points towards changing speeds as the particle accelerates or decelerates over time.

Characteristics of Parabolas on Graphs

A parabola is a type of curve commonly found in quadratic functions and can take different orientations based on the equation governing it. On a position-time graph, a parabola often signifies uniform acceleration, such as an object falling under gravity or an accelerating car.

A few key properties of parabolas include:

• They have a single vertex, which is the highest or lowest point of the curve.
• The direction of a parabola (upward or downward) indicates the kind of acceleration involved.
• If the parabola opens upwards, it suggests positive acceleration, while a downward-opening indicates negative acceleration.

Thus, recognizing the parabola on a position-time graph helps us understand the increasing or decreasing velocity of a particle and anticipate its motion.

Role of Tangent Lines in Understanding Motion

Tangent lines play a crucial role in reading curvilinear graphs, like those parabolic ones on position-time graphs. A tangent line is a straight line that touches a curve at only one point, representing the immediate direction in which the object is moving at that exact instant.

The slope of the tangent line reveals the instantaneous velocity of the particle. If the slope is steep, the velocity is higher, whether positive or negative. On a position-time graph, if you find a tangent line with a negative slope, it means the particle is moving in the opposite direction at that instant.

In our exercise, the tangent line's negative slope suggests immediate backward motion, indicative of a negative velocity.

Interpreting Negative Slopes and Instantaneous Velocity

Negative slopes on a position-time graph are indicative of specific movement characteristics. The slope of any line on this graph signifies the velocity of the particle. With a negative slope, the line trends downwards – this is crucial.

A downward-trending line means the position of the particle is decreasing over time; hence, the particle moves in a negative direction. This results in a negative instantaneous velocity, showing the particle is slowing down or moving backwards.

In practical terms, this negative velocity might be akin to a car reversing or an object moving back towards its starting point at that moment in time.