AP Physics C: Mechanics FRQ Practice

Practice FRQ 1 of 8

1. A ball is launched from ground level on a horizontal field with initial speed v0 at an angle θ above the horizontal, as shown in Figure 1. The ball lands on the ground at a horizontal distance R from the launch point. Air resistance is negligible, and the acceleration due to gravity has magnitude g. Let φ represent the angle that the velocity vector makes with the horizontal at the instant the ball lands.

Figure 1. Projectile launched from level ground with initial speed v₀ at angle θ; it lands a horizontal range R away with impact velocity making angle φ below the horizontal.

Single clean line diagram of projectile motion on level ground.

Page layout and reference frame:
- Use a wide landscape layout.
- Draw a straight horizontal ground line across the entire bottom of the figure.
- Place a Cartesian coordinate system at the launch point on the ground line:
- The origin is exactly on the ground line.
- The +x-axis is a solid arrow pointing horizontally to the right, lying along the ground line.
- The +y-axis is a solid arrow pointing straight upward.
- Label the axes with the visible text “x” at the right end of the horizontal axis arrow and “y” at the top of the vertical axis arrow.

Launch point and initial velocity:
- Mark the launch point at the origin with a small filled dot.
- From the origin, draw a bold velocity vector arrow labeled “v₀”.
- The vector points up and to the right (first quadrant).
- Its tail is exactly at the origin.
- Draw a thin angle arc at the origin between the +x-axis (reference line) and the v₀ vector.
- The arc is measured counterclockwise from the +x-axis up to the v₀ vector.
- Place the label “θ” next to this arc.

Trajectory and landing point:
- Draw a dotted parabolic trajectory starting at the origin and curving upward, then downward, returning to the ground line.
- The trajectory must intersect the ground line exactly twice: once at the origin (launch) and once at the landing point.
- Place the landing point on the ground line in the right half of the figure with a small filled dot.
- Label the horizontal distance from the launch point to the landing point as “R”:
- Draw a thin horizontal dimension line directly above the ground line, with arrowheads at both ends.
- The left arrowhead aligns vertically with the launch point; the right arrowhead aligns vertically with the landing point.
- Center the label “R” above this dimension line.

Landing velocity and impact angle:
- At the landing point, draw the instantaneous velocity vector as a bold arrow with its tail exactly at the landing dot.
- The vector points down and to the right (fourth quadrant), indicating downward vertical component at impact.
- Draw a thin angle arc at the landing point between the +x direction (use a short thin horizontal reference ray to the right starting at the landing point) and the landing velocity vector.
- The arc is measured clockwise downward from the +x reference ray to the landing velocity vector.
- Place the label “φ” next to this arc.

Styling and clarity requirements:
- All text labels must be clearly legible: “v₀”, “θ”, “R”, “φ”, “x”, “y”.
- Use dotted line style only for the projectile trajectory; all axes, vectors, and dimension lines are solid.
- Do not include any numeric tick marks or axis scales; this is a conceptual geometry diagram.
- Ensure θ is shown above the horizontal at launch, and φ is shown below the horizontal at landing.

Figure 2. Component-velocity grids at launch (t = 0) and landing (t = T).

Two side-by-side component-vector grids with identical scaling and formatting.

Overall layout:
- Use a wide landscape layout.
- Place two equal-size square grids separated by a clear vertical gap.
- Center the left grid in the left half of the page and the right grid in the right half of the page.

Grid formatting (applies to both grids identically):
- Each grid is a square with faint evenly spaced grid lines.
- Draw bold x- and y-axes through the center of each grid with arrowheads on the positive ends:
- +x is to the right; +y is upward.
- The axes intersect at the grid center; mark this intersection as the origin.
- Do not place any numeric tick labels.
- Label the top of each grid with a title centered above it:
- Above the left grid, the visible text: “At launch (t = 0)”
- Above the right grid, the visible text: “At landing (t = T)”

Left grid (launch components):
- Include one pre-drawn bold horizontal arrow representing the horizontal velocity component at launch:
- The arrow tail starts exactly at the origin of the left grid.
- The arrow points directly along the +x-axis.
- Place the label “vₓ” just above the arrow near its midpoint.
- No other arrows are pre-drawn in the left grid.

Right grid (landing components):
- No component arrows are pre-drawn in the right grid (blank aside from axes and grid).

Required proportionality cues for student-drawn arrows (must be implied by the grid scale):
- The two grids must have the same scale so that an arrow of a given length represents the same component magnitude in both grids.
- The pre-drawn launch vₓ arrow length defines the reference magnitude for vₓ; it should be long enough to be clearly measurable relative to the grid (spanning multiple small grid squares), but it must remain fully inside the grid boundaries.

Visual consistency:
- Use the same arrow thickness and arrowhead style for any component arrow shown.
- Keep all labels exactly as “vₓ” and “vᵧ” (with subscripts), matching AP Physics notation.
- Do not add any extra vectors, diagonal resultant arrows, or angle arcs in this figure; only component arrows belong on these grids.

i. The diagrams in Figure 2 can be used to represent the velocity components of the ball at launch and at landing. The horizontal velocity component at launch is shown in the left diagram.

Draw arrows on both grids to represent the remaining velocity components.

• Arrows should start at the origin.

• The length of the arrows should be proportional to the relative magnitudes of the components.

• Label each arrow with the appropriate velocity component symbol (

v_x

v_y

ii. Derive an expression for T, the time at which the ball lands. Express your answer in terms of

v_0

\theta

, g, 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 that relates φ to the launch angle θ. Express your answer in terms of θ and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.