Physics help - finding what slope means

[an_mui]

Suppose a student is given position-time data for a cart which was known to have an initial velocity of zero. if the student plots the fifth root of time (t^1/5) on the y axis, what variable should he plot on the x-axis so that the resulting graph is a straight line? how is the slope of the line related to the acceleration of the cart?
$$d = \frac{1}{2}at^2$$
$$d^\frac{1}{10} = (\frac{1}{2}a)^\frac{1}{10} t^\frac{1}{5}$$

this is what I've done so far... can anyone give me some hints as to what to do next?

 

[quantumdude]

Sorry for the late reply.

this is what I've done so far... can anyone give me some hints as to what to do next?

* Get $t^{1/5}$ by itself.
* Identify the independent variable (the one that is plotted on the x-axis).
* Identify the slope.

This can be done by comparing your equation to the equation $y=mx$ after you solve for $t^{1/5}$.

 

[quicknote]

Based on the given position-time data and the equation provided, it seems that the student should plot the fifth root of distance (d^1/5) on the y-axis and the square root of time (t^1/2) on the x-axis. This will result in a straight line with a slope of (1/2)a.

To understand why this is the case, let's break down the equation d = (1/2)at^2. This equation represents the distance (d) traveled by an object with an initial velocity of zero, as a function of time (t) and acceleration (a). When we take the fifth root of both sides, we get d^1/5 = ((1/2)a)^1/5 t^1/5. This means that if we plot the fifth root of distance on the y-axis and the fifth root of time on the x-axis, we will get a straight line with a slope of (1/2)a.

Now, let's look at the original question again. Instead of plotting the fifth root of distance, the student is asked to plot the fifth root of time on the y-axis. This means that the x-axis should represent a quantity that, when raised to the fifth power, gives us time. In other words, the x-axis should represent the fifth root of time, which is t^1/5. Therefore, the resulting graph will be a straight line with a slope of (1/2)a.

The slope of a position-time graph represents the velocity of the object. In this case, the slope of the graph is (1/2)a, which means that the velocity of the cart is changing by (1/2)a units every time the time is increased by one unit. This also means that the acceleration, represented by the coefficient a, is equal to twice the slope of the graph. In other words, the acceleration of the cart is (1/2)a units per unit time.

In summary, to plot a straight line with a slope that is related to the acceleration of the cart, the student should plot the fifth root of distance on the y-axis and the fifth root of time on the x-axis. This will result in a graph with a slope of (1/2)a, where a is the acceleration of the cart.