Learn Extracted exam questions AP Physics C: Mechanics 2015 Free Response
2015 Free Response
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A block of mass $m$ is projected up from the bottom of an inclined ramp with an initial velocity of magnitude $v_0$.
The ramp has negligible friction and makes an angle $\theta$ with the horizontal. A motion sensor aimed down the ramp is mounted at the top of the incline so that the positive direction is down the ramp. The block starts a distance $D$ from the motion sensor, as shown above. The block slides partway up the ramp, stops before reaching the sensor, and then slides back down.
[Diagram: An inclined ramp making angle $\theta$ with the horizontal. A Motion Sensor is mounted at the top of the incline, aimed down the ramp, with a $y$-axis pointing up-left from the sensor and an $x$-axis pointing along the ramp surface down and to the right. The point $x = 0$ is at the sensor (top of ramp); the point $x = D$ is at the bottom of the ramp where the block starts. A dashed line labeled $D$ spans from the sensor down to a small dashed box partway down the ramp. A block near the bottom of the ramp (at $x = D$) has an arrow labeled $\vec{v}_0$ pointing up the ramp (toward the sensor).]
Consider the motion of the block at some time $t$ after it has been projected up the ramp. Express your answers in terms of $m$, $D$, $v_0$, $t$, $\theta$ and physical constants, as appropriate.
Determine the acceleration $a$ of the block.
Determine an expression for the velocity $v$ of the block.
Determine an expression for the position $x$ of the block.
Derive an expression for the position $x_{min}$ of the block when it is closest to the motion sensor. Express your answer in terms of $m$, $D$, $v_0$, $\theta$, and physical constants, as appropriate.
On the axes provided below, sketch graphs of position $x$, velocity $v$, and acceleration $a$ as functions of time $t$ for the motion of the block while it goes up and back down the ramp. Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
[Three blank axes side by side, each with a horizontal $t$-axis and a vertical axis. The first is labeled $x$, the second labeled $v$, the third labeled $a$. All axes are unlabeled/blank grids for the student to sketch on — no data is plotted.]
After the block slides back down and leaves the bottom of the ramp, it slides on a horizontal surface with a coefficient of friction given by $\mu_k$. Derive an expression for the distance the block slides before stopping. Express your answer in terms of $m$, $D$, $v_0$, $\theta$, $\mu_k$, and physical constants, as appropriate.
Suppose the ramp now has friction. The same block is projected up with the same initial speed $v_0$ and comes back down the ramp. On the axes provided below, sketch a graph of the velocity $v$ as a function of time $t$ for the motion of the block while it goes up and back down the ramp, arriving at the bottom of the ramp at time $t_f$. Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
[Blank axes with vertical axis $v$ and horizontal axis $t$. Two tick marks are pre-labeled on the $t$-axis: $\dfrac{t_f}{2}$ and $t_f$. The graph itself is blank for the student to sketch.]
A small dart of mass 0.020 kg is launched at an angle of 30° above the horizontal with an initial speed of 10 m/s. At the moment it reaches the highest point in its path and is moving horizontally, it collides with and sticks to a wooden block of mass 0.10 kg that is suspended at the end of a massless string. The center of mass of the block is 1.2 m below the pivot point of the string. The block and dart then swing up until the string makes an angle $\theta$ with the vertical, as shown above. Air resistance is negligible.
[Diagram: A ceiling (hatched bar) with a string of length 1.2 m hanging straight down to a small square block. A dashed line at angle $\theta$ from the vertical string shows the block's swung-up position (dashed square), with $\theta$ marked at the pivot between the vertical string and the dashed string. Below, a dart is launched from the ground at 10 m/s at 30° above the horizontal (arrow labeled "10 m/s" with angle "30°" marked from the horizontal ground line); a dashed curved trajectory arcs up from the launch point to the block, reaching the block moving horizontally at the top of its arc. A horizontal distance $d$ is marked along the ground from the dart's launch point to the point on the floor directly below the block.]
Determine the speed of the dart just before it strikes the block.
Calculate the horizontal distance $d$ between the launching point of the dart and a point on the floor directly below the block.
Calculate the speed of the block just after the dart strikes.
Calculate the angle $\theta$ through which the dart and block on the string will rise before coming momentarily to rest.
The block then continues to swing as a simple pendulum. Calculate the time between when the dart collides with the block and when the block first returns to its original position.
In a second experiment, a dart with more mass is launched at the same speed and angle. The dart collides with and sticks to the same wooden block.
Would the angle $\theta$ that the dart and block swing to increase, decrease, or stay the same?
_____ Increase _____ Decrease _____ Stay the same
Justify your answer.
Would the period of oscillation after the collision increase, decrease, or stay the same?
_____ Increase _____ Decrease _____ Stay the same
Justify your answer.
A uniform, thin rod of length $L$ and mass $M$ is allowed to pivot about its end, as shown in the figure above.
[Diagram: A horizontal thin rod of length $L$ pivoted at its left end (marked "Pivot Point" with a pivot symbol), extending to the right to a point labeled $M$ (the mass) at its far end. The length $L$ is marked with a double-headed arrow spanning the rod above it.]
Using integral calculus, derive the rotational inertia for the rod around its end to show that it is $ML^2/3$.
[Diagram: A ceiling mount at the top left holding a horizontal rod via a pivot, with the rod extending to the right and labeled $A$ at its free end. A curved arrow sweeps clockwise from the horizontal rod position down to a vertical dashed line, showing the rod falling from horizontal to vertical. The vertical dashed position is labeled $B$ at its bottom end.]
The rod is fixed at one end and allowed to fall from the horizontal position $A$ through the vertical position $B$.
Derive an expression for the velocity of the free end of the rod at position $B$. Express your answer in terms of $M$, $L$, and physical constants, as appropriate.
An experiment is designed to test the validity of the expression found in part (b). A student uses rods of various lengths that all have a uniform mass distribution. The student releases each of the rods from the horizontal position $A$ and uses photogates to measure the velocity of the free end at position $B$. The data are recorded below.
| Length (m) | 0.25 | 0.50 | 0.75 | 1.00 | 1.25 | 1.50 |
|---|---|---|---|---|---|---|
| Velocity (m/s) | 2.7 | 3.8 | 4.6 | 5.2 | 5.8 | 6.3 |
[Two additional blank rows below the data table, with the same six columns, for the student to fill in derived quantities.]
Indicate below which quantities should be graphed to yield a straight line whose slope could be used to calculate a numerical value for the acceleration due to gravity $g$.
Horizontal axis: ________________
Vertical axis: ________________
Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not given. Label each row you use and include units.
Plot the straight line data points on the grid below. Clearly scale and label all axes, including units as appropriate. Draw a straight line that best represents the data.
[A blank grid divided into a 5×6 array of larger cells, each subdivided into finer dashed gridlines, for plotting data points and drawing a best-fit line. No axes are labeled and no data is plotted.]
Using your straight line, determine an experimental value for $g$.
Describe two ways in which the effects of air resistance could be reduced.