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The slope of any line is rise/run, or change in y divided by change in x.
On a distance-time curve, time is the variable on the x axis, and distance is the variable on the y axis. This means that when a tangent is drawn at any point on the curve, its slope becomes change in distance divided by change in time, for example, m/s, km/h, etc. These units align with the units for velocity, and therefore the slope of the tangent line on a distance-time curve is the velocity.
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What physical property does the slope of a velocity time graph represent?
velocity.
What is instantaneous slope?
The instantaneous slope of a curve is the slope of that curve at a single point. In calculus, this is called the derivative. It also might be called the line tangent to the curve at a point. If you imagine an arbitrary curve (just any curve) with two points on it (point P and point Q), the slope between P and Q is the slope of the line connecting those two points. This is called a secant line. If you keep P where it is and slide Q closer and closer to P along the curve, the secant line will change slope as it gets smaller and smaller. When Q gets extremely close to P (so that there is an infinitesimal space between P and Q), then the slope of the secant line approximates the slope at P. When we take the limit of that tiny distance as it approaches zero (meaning we make the space disappear) we get the slope of the curve at P. This is the instantaneous slope or the derivative of the curve at P. Mathematically, we say that the slope at P = limh→0 [f(x+h) - f(x)]÷h = df/dx, where h is the distance between P and Q, f(x) is the position of P, f(x+h) is the position of Q, and df/dx is the derivative of the curve with respect to x. The formula above is a specific case where the derivative is in terms of x and we're dealing with two dimensions. In physics, the instantaneous slope (derivative) of a position function is velocity, the derivative of velocity is acceleration, and the derivative of acceleration is jerk.
What does the slope of the curve on a velocity time graph represent?
The rate of Change in acceleration.
What is a derivative in calculus?
When you take the derivative of a function, you are seeking a variation of that function that provides you with the slope of the tangent (instantaneous slope) at any value of (x). For example, the derivative of the function f(x)=x^2 is f'(x)=2x. Notice that the derivative is denoted by the apostrophe inside the f and (x). Also note that at x=0, f'(x)=0, which means that at x=0 the slope of the tangent is zero, which is correct for the function y=x^2.
How do you find the equation of a tangent line to the graph of fx equals x cubed and x equals -2?
Calculate the derivative of the function.Use the derivative to calculate the slope at the specified point.Calculate the y-coordinate for the point.Use the formula for a line that has a specified slope and passes through a specified point.
Does the slope of a position time graph represent average velocity or rate of change of velocity?
The slope of a position-time graph represents the average velocity of an object. It does not represent the rate of change of velocity, which would be represented by the slope of a velocity-time graph.
What does the slope of the tangent to the curve on a velocity-time graph measure?
The slope of the tangent to the curve on a velocity-time graph represents the acceleration of an object. Positive slope indicates acceleration in the positive direction, negative slope indicates acceleration in the negative direction, and zero slope indicates constant velocity.
Is there a relationship between the sign positive or negative of the slope of a line and the angle the line makes with the x axis?
yes, the slope of the line is the tangent of the angle
Is Average velocity the slope of the line tangent to the curve on a velocity time graph at a specific instant of time?
No, average velocity is the total displacement divided by the total time taken. The slope of the tangent to the curve on a velocity-time graph at a specific instant of time gives the instantaneous velocity at that moment, not the average velocity.
Short-lived velocity defined as the slope of the tangent line at a given point on a curve?
instantaneous
How do you get the slope of a line on a graph?
Take a tangent at the point where you want the slope. Then the slope of the graph at that point is the slope of the tangent, which is found by taking another point on the tangent and then taking the change in y between the two points and divid it by the change in x.
What does the slope of a tangent to the curve of a velocity-time graph measure?
The slope of a tangent to the curve of a velocity-time graph represents the acceleration of an object at that specific instant in time. A steeper slope indicates a greater acceleration, while a flatter slope indicates a smaller acceleration.
What is the slope of the line tangent to the curve on a position-time graph at a specific time?
The slope of the line tangent to the curve on a position-time graph at a specific time represents the velocity of the object at that particular moment. It indicates how fast the object is moving at that instant.
What does a tangent to a velocity-time graph measure?
A tangent to a velocity-time graph represents the instantaneous acceleration of an object at that specific moment in time. It shows how the velocity is changing at that particular point.
How would you obtain the acceleration of an object from a velocity time graph?
The average acceleration can be obtained by finding the slope of the graph. The instantaneous acceleration is found by drawing a tangent to a particular point on the graph (instant) and finding the slope of than tangent.
How is instantaneous acceleration related to a velocity-time graph?
Instantaneous acceleration at any point on a velocity-time graph can be determined by calculating the slope of the tangent line at that specific point. A steeper slope represents a higher acceleration, while a shallower slope indicates a lower acceleration.
What does the slope of distance versus time graph represent physically?
The slope of a distance versus time graph represents the speed or velocity of an object. A steeper slope indicates a higher speed, while a gentler slope indicates a slower speed. If the slope is negative, it means the object is moving in the opposite direction.
