Question

a cart is released from rest from the top of an inclined ramp. ignoring friction, which of the following is the best representation of the x - position vs. time graph for the cart as it rolls down the ramp. assume x = 0 is at the top of the ramp, t = 0 occurs at the moment the cart is released, and the +x axis points directly down the ramp.

Explanation:

Step1: Recall motion - equation for constant - acceleration

The motion of the cart on the inclined ramp is a one - dimensional motion with constant acceleration. The position - time equation for an object starting from rest ($v_0 = 0$) with constant acceleration $a$ is $x=x_0 + v_0t+\frac{1}{2}at^{2}$. Since $x_0 = 0$ and $v_0 = 0$, the equation simplifies to $x=\frac{1}{2}at^{2}$, which is a quadratic function of the form $y = Ax^{2}$ ($A=\frac{a}{2}$ in our case).

Step2: Analyze the shape of the quadratic function

The graph of a quadratic function $y = Ax^{2}$ with $A>0$ (and here $a>0$ as the cart is accelerating down the ramp) is a parabola opening upwards.

Answer:

The first graph (the one with a curve opening upwards) is the best representation.