Home / AP® Exam / AP Physics C Mechanics / Exam Style Questions MCQs and FRQs / Motion in Two or Three Dimensions FRQ

AP Physics C Mechanics - 1.5 Motion in Two or Three Dimensions- Exam Style questions- FRQs

Motion in Two or Three Dimensions AP  Physics C Mechanics FRQ

Unit: 1. Kinematics 

Weightage : 10-15%

AP Physics C Mechanics Exam Style Questions – All Topics

Question

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 \(3R\) 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.
 
(a) On the dot below, which represents the block, draw and label the forces (not components) that act on the block while still in contact with the track at point \(B\). Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. Justify your choice of vectors.
(b)
i. Derive an expression for the speed \(v\) of the block at point \(B\).
ii. Derive an expression for the magnitude of the net force \(F\) on the block at point \(B\).
(c) Derive an expression for the minimum value of \(\Delta x_{min}\) required in order for the block to maintain contact with the track through point \(B\).
The procedure is repeated several times with the distance \(\Delta x > \Delta x_{min}\).
(d) Calculate the distance \(D\) that the block travels.
(e) The graph below shows the best-fit line drawn by the students through their data of \(D\) as a function of \(\Delta x\).
i. Explain why there are no data for section I of the graph.
ii. Explain the reason for the shape and minimum value of section II on the graph.

Most-appropriate topic codes (AP Physics C: Mechanics):

• Topic \(2.2\) — Forces and Free-Body Diagrams (Part \( \mathrm{a} \))
• Topic \(3.4\) — Conservation of Energy (Part \( \mathrm{b} \))
• Topic \(2.10\) — Circular Motion (Parts \( \mathrm{b} \), \( \mathrm{c} \))
• Topic \(1.5\) — Motion in Two or Three Dimensions (Part \( \mathrm{d} \))
▶️ Answer/Explanation

(a)


The force \(F_g\) represents the weight of the block and always points downward.
The force \(F_N\) represents the force the track exerts on the block to keep it moving in a circular path and points perpendicular to the surface of the track.

(b)(i)
Using conservation of energy:
\(U_{s1} + K_1 = U_{g2} + K_2\)
\(\dfrac{1}{2}k(\Delta x)^2 + 0 = mg(3R) + \dfrac{1}{2}mv_B^2\)
\(v_B = \sqrt{\dfrac{k}{m}(\Delta x)^2 – 6gR}\)

(b)(ii)
Relating centripetal force to speed from part (b)(i):
\(F_C = \dfrac{mv_B^2}{R}\)
\(F_C = \dfrac{m}{R}\left(\dfrac{k}{m}(\Delta x)^2 – 6gR\right)\)
\(F_{net} = \dfrac{k(\Delta x)^2}{R} – 6mg\)

(c)
Relating the net force to the diagram from part (a) and setting the normal force equal to zero:
\(\dfrac{k(\Delta x)^2}{R} – 6mg = mg\)
\(\dfrac{k(\Delta x)^2}{R} = 7mg\)
\(\Delta x_{min} = \sqrt{\dfrac{7mgR}{k}}\)

(d)
Relating the height of the fall to the time of fall:
\(H = \dfrac{1}{2}gt^2\)
\(t = \sqrt{\dfrac{8R}{g}}\)
Using the equation for constant horizontal velocity:
\(D = v_x t\)
\(D = \left(\sqrt{\dfrac{k}{m}(\Delta x)^2 – 6gR}\right)\left(\sqrt{\dfrac{8R}{g}}\right)\)

(e)(i)
The block needs a minimum speed to make it through point B on the track; thus, the horizontal line segment represents compressions of the spring for which the block does not make it to point B.

(e)(ii)
From the equation in part (d), the horizontal distance traveled by the block is directly proportional to the compression of the spring; thus, the graph would be a straight line.
The minimum value is the distance traveled when the compression of the spring generates the minimum speed needed to reach point B on the track.

More AP Physics C Mechanics FRQ Questions..
Scroll to Top