Question

The slope of the tangent drawn on a position-time graph at any instant is equal to the instantaneous ____.

• A. acceleration
• B. force
• C. velocity
• D. momentum
• E. impulse

Hint:

To remember the relationships: The slope of Position-Time gives Velocity, and the slope of Velocity-Time gives Acceleration.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
In a position-time ($x-t$) graph, the vertical axis represents position and the horizontal axis represents time. The slope of a line is defined as the change in the vertical coordinate divided by the change in the horizontal coordinate.

Step 2: Key Formula or Approach
Slope ($m$) = $\frac{dx}{dt}$. In physics, the rate of change of position with respect to time is defined as velocity ($v$).

Step 3: Detailed Explanation
1. For any graph, the slope at a specific point (tangent) represents the instantaneous rate of change.
2. On an $x-t$ graph, a steeper slope indicates a higher speed, while a horizontal line (slope = 0) indicates the object is at rest.
3. Mathematically, $v_{inst} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$, which is exactly the definition of the tangent's slope.

Step 4: Final Answer
The slope of the tangent on a position-time graph is equal to the instantaneous velocity.