⚙️AP Physics C: Mechanics
1. A small object is launched from ground level at time t = 0 with initial velocity components v_x0 in the horizontal direction and v_y0 in the vertical direction. The object experiences a horizontal acceleration a_x(t) = kt, where k is a positive constant, and a vertical acceleration a_y(t) = -g + bt^2, where b is a positive constant and g is the acceleration due to gravity. The object reaches its maximum height at time t = t_m and returns to ground level at time t = t_e.
Figure 1: Horizontal and vertical velocity components versus time (from launch at t = 0 to landing at t = t_e)
On the axes in Figure 1, sketch the horizontal velocity component v_x and the vertical velocity component v_y as functions of time from t = 0 to t = t_e. Clearly indicate the initial values v_x0 and v_y0, and label the time t_m on your graphs where appropriate.
Your sketches should show the general shape and curvature of each velocity component.
The relative magnitudes and signs should be consistent with the given accelerations.
Derive an expression for the vertical velocity v_y as a function of time t. Express your answer in terms of v_y0, g, b, t, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Figure 2: Ground reference frame (x, y) and observer reference frame (x′, y′) with observer moving at constant speed v_obs in +x
Derive an expression for v_x0_prime, the initial horizontal velocity component of the object in the observer's reference frame. Express your answer in terms of v_x0, v_obs, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. An observer moves with constant velocity v_obs = v_x0 in the +x direction relative to the ground frame, as shown in Figure 2. In the observer's reference frame, the object has initial horizontal velocity component v_x0_prime at t = 0.
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