AP Physics C: Mechanics FRQ Practice

⚙️AP Physics C: Mechanics

Practice FRQ 1 of 81/8

1. A model rocket of mass m is launched vertically upward from the ground. The rocket engine produces a time-varying net upward acceleration (accounting for gravity) given by

a(t) = a_0(1 - \frac{t}{T})

for

0 ≤ t ≤ T

, where

a_0

is the initial net acceleration and T is the time when the engine shuts off. At time

t = T

, the engine shuts off and the rocket continues upward, subject only to gravity. Let the upward direction be positive, and let

y = 0

be the position of the rocket at

t = 0

. Figure 1 shows the rocket at three stages of motion.

Figure 1: Model rocket at launch, engine shutoff, and maximum height (upward is +y)

A clean black-and-white physics schematic (no perspective), oriented vertically.

Layout and coordinate reference:
- Draw a vertical reference line centered in the diagram representing the rocket’s line of motion.
- At the left side of this line, draw a y-axis: a vertical axis with an arrow at the top labeled '+y'.
- Place the label 'y = 0' at ground level, aligned with the lowest rocket position.

Ground and launch state (bottom stage):
- Draw a thin horizontal ground line across the lower part of the diagram.
- Place the first rocket directly on the ground line, centered on the vertical reference line.
- Immediately next to this bottom rocket (slightly to the right of the rocket body), place the text label 't = 0'.
- Next to the same bottom rocket, place the text label 'y = 0'.
- Draw a single acceleration arrow originating at the rocket’s center and pointing straight upward. Make this arrow the longest arrow in the entire figure.
- Label this arrow 'a₀' with the text placed just to the right of the arrow shaft, vertically centered on the arrow.

Engine shutoff state (middle stage):
- Place the second rocket on the same vertical reference line, vertically above the bottom rocket by a clear, moderate spacing (so it is visually distinct but not near the top).
- To the right of this middle rocket, place the label 't = T'.
- Next to the middle rocket, indicate its height from the ground by writing 'h₁' as a label to the right of the rocket, with a short horizontal leader line pointing to the rocket.
- Draw an upward velocity arrow originating at the center of the middle rocket and pointing straight up.
- Make this velocity arrow shorter than the bottom acceleration arrow.
- Label this velocity arrow 'v₁' with the text placed just to the right of the arrow shaft.

Maximum height state (top stage):
- Place the third rocket on the same vertical reference line, near the top of the diagram and clearly above the middle rocket.
- Next to this top rocket, place the label 'h_max' (with 'max' as a subscript).
- Also place the label 'v = 0' next to the top rocket (to the right of the rocket body), indicating the rocket is momentarily at rest.
- Do NOT draw any velocity arrow at the top stage.

Relative consistency requirements:
- All three rockets are identical in size and orientation (pointing upward).
- The three rockets are vertically aligned on one straight line.
- The only arrow at launch is the upward acceleration arrow labeled 'a₀'.
- The only arrow at engine shutoff is the upward velocity arrow labeled 'v₁'.
- At maximum height, there is no motion arrow, only the text 'v = 0'.

Text and styling:
- Use solid black lines, medium stroke for rockets and arrows, thin stroke for axes and ground.
- No numeric values appear in Figure 1 other than the symbolic labels listed above.

Figure 2: Blank axes for velocity–time and position–time sketches (engine shuts off at T = 2.0 s)

Two separate coordinate grids placed side-by-side with equal height and equal visual weight, intended for student sketches.

Overall layout:
- Left panel: velocity versus time graph.
- Right panel: position versus time graph.
- The two panels are separated by a clear vertical gap so the axes and labels do not overlap.
- Both panels include light square grid lines aligned with tick marks.

LEFT PANEL (Velocity–time):
Axes:
- Horizontal axis labeled 'Time (s)'.
- Time axis runs from 0 to 5.
- Tick marks and numeric labels at every 1 second: 0, 1, 2, 3, 4, 5.
- Vertical axis labeled 'Velocity (m/s)'.
- Velocity axis runs from 0 to 50.
- Tick marks and numeric labels at every 10 m/s: 0, 10, 20, 30, 40, 50.
- The origin at the axes intersection is explicitly labeled '0'.
- Arrowhead at the positive end of the time axis (to the right).
- Arrowhead at the positive end of the velocity axis (upward).
Grid:
- Vertical grid lines at every 1 second matching the time ticks.
- Horizontal grid lines at every 10 m/s matching the velocity ticks.
Engine shutoff marker (must be present even though curves are not drawn):
- Draw a thin vertical dashed line exactly at the time tick labeled '2'.
- Place the label 'T = 2.0 s' above the top of this dashed line, inside the plotting area near the top edge.

RIGHT PANEL (Position–time):
Axes:
- Horizontal axis labeled 'Time (s)'.
- Time axis runs from 0 to 5.
- Tick marks and numeric labels at every 1 second: 0, 1, 2, 3, 4, 5.
- Vertical axis labeled 'Position (m)'.
- Position axis runs from 0 to 150.
- Tick marks and numeric labels at every 30 m: 0, 30, 60, 90, 120, 150.
- The origin at the axes intersection is explicitly labeled '0'.
- Arrowhead at the positive end of the time axis (to the right).
- Arrowhead at the positive end of the position axis (upward).
Grid:
- Vertical grid lines at every 1 second matching the time ticks.
- Horizontal grid lines at every 30 m matching the position ticks.
Engine shutoff marker (must be present even though curves are not drawn):
- Draw a thin vertical dashed line exactly at the time tick labeled '2'.
- Place the label 'T = 2.0 s' above the top of this dashed line, inside the plotting area near the top edge.

Style constraints:
- No curves are pre-drawn on either panel.
- No titles other than the axis labels.
- All numbers listed above must appear exactly as tick labels.

i. On the axes in Figure 2, sketch graphs of the rocket's velocity versus time and position versus time for the entire motion from launch until the rocket reaches its maximum height. The engine shuts off at time

T = 2.0

s. Clearly indicate on your graphs:

• The time T when the engine shuts off

• The qualitative shape of each curve in both the engine-on and engine-off phases

• The maximum height on the position graph

ii. Derive an expression for the rocket's velocity

v_1

at time

t = T

when the engine shuts off. Express your answer in terms of

a_0

, T, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

B. Derive an expression for the maximum height

h_{max}

reached by the rocket, measured from the ground. Express your answer in terms of

a_0

, T, g, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.