Kinematics and Motion

Consider a projectile launched with initial velocity $v_0$ at an angle $\theta$ with respect to the horizontal. Accounting for air resistance, which is proportional to the square of the velocity, what is the qualitative effect on the range and maximum height compared to the ideal projectile motion without air resistance?

A particle's position is given by $r(t) = (At^3 + Bt)i + (Ct^2 + Dt^4)j$, where A, B, C, and D are non-zero constants. Determine the conditions under which the particle's acceleration will have only a j-component at a specific time $t_0$.

A block of mass $m_1$ rests on a horizontal, frictionless surface and is connected to a hanging mass $m_2$ by a massless string over a massless, frictionless pulley. Determine the Lagrangian for this system in terms of the horizontal displacement $x$ of $m_1$.

Consider an Atwood machine with two masses, $m_1$ and $m_2$, connected by a massless string over a pulley with moment of inertia $I$ and radius $R$. Derive the equation of motion for the system using the Euler-Lagrange equation, expressed in terms of the angular acceleration $\alpha$ of the pulley.

A projectile is launched on a planet with no atmosphere but a non-uniform gravitational field described by $g(y) = g_0e^{-ky}$, where $g_0$ is the gravitational acceleration at the surface (y=0) and k is a positive constant. Determine a modified expression for the maximum height reached by the projectile, assuming the initial vertical velocity component is v_y0.

Two blocks, $m_1$ and $m_2$ ($m_1 > m_2$), are connected by a string that passes over a pulley. Considering the effects of a non-ideal, massive pulley with radius $R$ and moment of inertia $I$, along with friction in the pulley's axle (modeled as a torque $\tau$), derive the expression for the acceleration of the masses.

A vehicle moving at an initial velocity $v_0$ applies its brakes, resulting in a deceleration that increases linearly with time, described by $a(t) = -kt$, where k is a positive constant. Determine the time at which the vehicle comes to a complete stop.

A projectile is launched on level ground with an initial velocity $v_0$ at an angle $\theta$. Determine an expression for the time at which the projectile's speed is minimized during its flight.

Consider a modified Atwood machine where the pulley has a non-negligible mass and radius, leading to rotational inertia $I$. If the two masses, $m_1$ and $m_2$, are released from rest, and the string does not slip on the pulley, determine the ratio of the tensions $T_1/T_2$ in the string segments connected to $m_1$ and $m_2$ respectively.

A ball is thrown vertically upwards with an initial velocity $v_0$ in a medium where the drag force is proportional to the velocity, i.e., $F_d = -bv$, where b is a positive constant. Determine the time it takes for the ball to reach its maximum height.