[Physics] Motion on an incline: is the displacement the length of the flat horizontal leg of the ramp

kinematics

This may seem like a silly question, so pardon in advance. I was working through some practice 2D Kinematic word problems, and I was wondering whether my approach is wrong.

I know how to calculate the acceleration in the x-direction and the velocity in the x-direction, but up until now, I've been assuming that the displacement ($x_f-x_i$) is equal to the length of the horizontal leg of the triangle formed by the ramp. Is it in fact the length of the hypotenuse? This doesn't seem right to me. The reason I'm confused is because of this practice problem. My intuition tells me that they solved this problem incorrectly.

Best Answer

The length of the hypotenuse is a displacement vector representing the sum of the displacement vector in the x direction and the displacement vector in the y direction. The distance traveled in this case is also the length of the hypotenuse, even though it is a scalar rather than a vector.

Acceleration, velocity, and displacement are vectors. Each of them is composed of a y vector and an x vector. When an object rolls down an incline, the magnitude of displacement is the length of the incline, which can be represented as the sum of the x-displacement and the y-displacement vectors, which can be computed as the hypotenuse of the right triangle they form.