Answer is: option1

0

Solution:

We need to determine the number of values of \( t \) in the interval \( 0 \leq t \leq 2\pi \) where the two particles have the same velocity. The velocities are found by differentiating their position functions.

The position functions given are: \[ x_1 = \cos t, \quad x_2 = e^{(t-3)/2} - 0.75 \] Velocity functions: \[ v_1 = \frac{dx_1}{dt} = -\sin t \] \[ v_2 = \frac{dx_2}{dt} = \frac{1}{2} e^{(t-3)/2} \] Solve for \( t \) where \( v_1 = v_2 \): \[ -\sin t = \frac{1}{2} e^{(t-3)/2} \] Number of Solutions in \( [0, 2\pi] \) is zero.