⚙️AP Physics C: Mechanics
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 2. Component-velocity grids at launch (t = 0) and landing (t = T).
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 ( or ).
Derive an expression for T, the time at which the ball lands. Express your answer in terms of , , g, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
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.
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. Consider the angle φ that the velocity vector makes with the horizontal at the instant the ball lands, as shown in Figure 1.
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