AP Physics C Mechanics 1.5 2D & 3D Motion Overview

Analyze motion in two or three dimensions by resolving vectors into components and applying calculus-based relationships to each coordinate direction.

[Maximum number: 2]

A block of mass m is placed on top of an ideal spring of spring constant k. The block is pushed against the spring, compressing the spring a distance $\Delta x$ . The block is released from rest, leaves the spring at the position shown in the figure, travels upward, and enters a track with a constant radius of curvature R that has negligible friction. The block enters the track at point A, maintains contact with the track, and exits horizontally at point B, a distance 3 R above the point the block was released. The block then falls to the ground and lands a horizontal distance D from the end of the track. Express all algebraic answers in terms of $m, k, \Delta x, R$ , and physical constants, as appropriate. The size of the block is much smaller than the radius of curvature of the track.

Calculate the distance D that the block travels.

For correctly relating the height of fall to the time of fall
$y=y_{0}+v_{o y} t+\frac{1}{2} a_{y} t^{2} \therefore H=0+0+\frac{1}{2} g t^{2} \therefore t=\sqrt{\frac{(2)(4 R)}{g}}=\sqrt{\frac{8 R}{g}}$

1 point

For correctly substituting into the equation for constant velocity consistent with part (b)(i)
$D=v_{x} t=\left(\sqrt{\frac{k}{m}(\Delta x)^{2}-6 g R}\right)\left(\sqrt{\frac{8 R}{g}}\right)$

1 point

Total for part (d)

2 points

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Covered syllabus points

1.5 Motion in Two or Three Dimensions question 1