AP Physics C Mechanics - 1.3 Representing Motion- Exam Style questions- FRQs
Representing Motion AP Physics C Mechanics FRQ
Unit: 1. Kinematics
Weightage : 10-15%
Question
Blocks A and B of masses \(2m\) and \(m\), respectively, are arranged in a setup consisting of a ramp that makes an angle \(\theta\) with a smooth horizontal table and an ideal spring of spring constant \(k\) fixed to a wall, as shown. Block A is held at rest a distance \(D\) up the ramp, and Block B is at rest on the horizontal table. The coefficient of kinetic friction between Block A and the rough ramp is \(\mu\) in the region of length \(D\), and there is negligible friction between the blocks and the smooth table.
At time \(t=0\), Block A is located at horizontal position \(x=0\) and is released from rest. After the block is released, the following occurs.
• At time \(t=t_{1}\) Block A has traveled a distance \(D\) down the ramp, has transitioned to the table, and is moving with speed \(v\) at \(x=x_{1}\).
• At time \(t=t_{2}\) Block A is at \(x=x_{2}\) when it collides with and sticks to Block B.
• At time \(t=t_{3}\), the combined blocks A and B are at \(x=x_{3}\) when they collide with and stick to the spring in its equilibrium position.
• At time \(t=t_{4}\) the combined blocks A and B are instantaneously at rest and the spring is compressed a distance \(x_{c}\) from its equilibrium position.
• At time \(t=t_{2}\) Block A is at \(x=x_{2}\) when it collides with and sticks to Block B.
• At time \(t=t_{3}\), the combined blocks A and B are at \(x=x_{3}\) when they collide with and stick to the spring in its equilibrium position.
• At time \(t=t_{4}\) the combined blocks A and B are instantaneously at rest and the spring is compressed a distance \(x_{c}\) from its equilibrium position.
(a) For parts (a)(i) and (a)(ii), express your answer in terms of \(m\), \(\theta\), \(D\), \(\mu\), \(x_{c}\), and physical constants, as appropriate.
i. Derive an expression for the speed \(v\) of Block A at time \(t_{1}\).
ii. Derive an expression for the spring constant \(k\) of the spring.
ii. Derive an expression for the spring constant \(k\) of the spring.
(b)
i. On the following axes, sketch a graph of the magnitude of the momentum \(p_{A}\) of Block A as a function of time from \(t=0\) to \(t_{4}\).
ii. Use principles of forces to justify the graph drawn in part (b)(i) for the time interval \(t=t_{3}\) to \(t=t_{4}\). Explicitly reference features of the shape of the graph you drew in part (b)(i).
For times \(t>t_{4}\) the two-block-spring system oscillates with period \(T_{O}\). The procedure is then repeated using a new ramp, where there is negligible friction between Block A and the ramp.
(c) Indicate how the new period of oscillation \(T_{N}\) in the procedure that uses the new ramp compares with the period of oscillation \(T_{O}\) from the original procedure.
_____ \(T_{N} > T_{O}\) _____ \(T_{N} < T_{O}\) _____ \(T_{N} = T_{O}\)
Briefly justify your answer.
Most-appropriate topic codes (AP Physics C: Mechanics):
• Topic \(1.3\) — Representing Motion (Part \(\mathrm{b}\))
• Topic \(2.5\) — Newton’s Second Law (Part \(\mathrm{b}\))
• Topic \(3.4\) — Conservation of Energy (Part \(\mathrm{a}\))
• Topic \(4.3\) — Conservation of Linear Momentum (Part \(\mathrm{a}\))
• Topic \(4.4\) — Elastic and Inelastic Collisions (Part \(\mathrm{a}\))
• Topic \(7.2\) — Frequency and Period of SHM (Part \(\mathrm{c}\))
• Topic \(2.5\) — Newton’s Second Law (Part \(\mathrm{b}\))
• Topic \(3.4\) — Conservation of Energy (Part \(\mathrm{a}\))
• Topic \(4.3\) — Conservation of Linear Momentum (Part \(\mathrm{a}\))
• Topic \(4.4\) — Elastic and Inelastic Collisions (Part \(\mathrm{a}\))
• Topic \(7.2\) — Frequency and Period of SHM (Part \(\mathrm{c}\))
▶️ Answer/Explanation
(a)(i)
By applying the conservation of energy to Block A as it slides down the rough incline, the initial gravitational potential energy is converted into kinetic energy and work done against friction.
\(E_{initial} = E_{final}\)
\(U_{g} – \Delta E_{friction} = K\)
\(m_{A} g D \sin\theta – \mu m_{A} g D \cos\theta = \frac{1}{2}m_{A} v^2\)
\((2m) g D \sin\theta – \mu (2m) g D \cos\theta = \frac{1}{2}(2m) v^2\)
Solving for \(v\), we get:
\( \boxed{v = \sqrt{2gD(\sin\theta – \mu\cos\theta)}} \)
(a)(ii)
When Block A collides perfectly inelastically with Block B, linear momentum is conserved:
\(p_{before} = p_{after}\)
\(2mv = (2m+m)v_{AB}\)
\(v_{AB} = \frac{2}{3}v\)
Next, we use conservation of mechanical energy as the combined blocks compress the spring to a stop:
\(K_{after\_collision} = U_{s,max}\)
\(\frac{1}{2}(3m)v_{AB}^2 = \frac{1}{2}kx_{c}^2\)
Substituting our previous expression for \(v\) and solving for the spring constant \(k\):
\((3m) \left(\frac{2}{3} \sqrt{2gD(\sin\theta-\mu\cos\theta)}\right)^2 = k x_{c}^2\)
\( \boxed{k = \frac{8mgD(\sin\theta-\mu\cos\theta)}{3x_{c}^2}} \)
(b)(i)
Block A’s momentum \(p_{A}\) increases linearly from \(0\) to \(t_{1}\) as it accelerates down the ramp. From \(t_{1}\) to \(t_{2}\), \(p_{A}\) is represented by a constant horizontal line while moving on the smooth table. At \(t_{2}\), the inelastic collision causes an instant drop to a lower constant value until \(t_{3}\). Finally, from \(t_{3}\) to \(t_{4}\), the spring force slows it down, so the graph is drawn as a concave down curve reaching zero at \(t_{4}\).
(b)(ii)
Between \(t_{3}\) and \(t_{4}\), the spring exerts a restoring force in the opposite direction of motion. Because the magnitude of the spring force increases as it compresses (\(F = -kx\)), the deceleration of Block A increases over time, resulting in a momentum graph that becomes increasingly steeper (concave down) until the block completely stops.
(c)
Correct Selection: \( \boxed{T_{N} = T_{O}} \)
The period of a mass-spring oscillating system depends strictly on the oscillating mass and the spring constant (\(T = 2\pi\sqrt{m/k}\)). Since repeating the experiment on a frictionless ramp only increases the block’s initial speed and the spring’s maximum compression distance (amplitude) without changing the system’s mass or spring constant, the new period \(T_{N}\) will remain identical to the original period \(T_{O}\).