12.- The acceleration of an object that travels in the $x$-direction is given by $a(x)=4x-2\ \text{m/s}^2$. If its velocity is $v_0=10\ \text{m/s}$ at $x_0=0\ \text{m}$, find its velocity for any position $x$.

13.- A particle describes a movement in the $xy$-plane so that the Cartesian components of its velocity vector, expressed in the International System, are $v_x(t)=4t^3+4t$ and $v_y(t)=4t$. If the particle is at the point of coordinates $(1,2)$ at $t_0=0\ \text{s}$, find the Cartesian equation of its trajectory.

14.- A particle describes a trajectory in the $xy$-plane so that the parametric equations describing its motion are $x(t)=pt$, $y(t)=\frac{1}{2}pt^2$, where $p$ is a constant. Find: (a) The Cartesian components of the velocity and acceleration vectors as a function of time, and their magnitudes. (b) The intrinsic components of the acceleration. (c) The radius of curvature of the trajectory.

15.- An object is thrown vertically upward with a speed of $98\ \text{m/s}$, from the edge of the roof of a $100\ \text{m}$ tall building. The object doesn't hit the building on its back down and lands in the street below. Find: (a) The maximum height reached by the object as measured from the ground level. (b) The time when the object passes through its launching point. (c) The velocity of the object just before it reaches the ground. (d) The time elapses when the object reaches the ground.

16.- A rock is thrown vertically upward from the top of a tower with an initial velocity of $15\ \text{m/s}$. If the origin of coordinates is fixed at the launching point of the rock, find: (a) The position and the velocity of the rock after $1\ \text{s}$ and $4\ \text{s}$ since it was launched. (b) The velocity of the rock when it is $8\ \text{m}$ above the launching point. (c) The time elapsed until the rock passes again through the launching point. Assume $g=10\ \text{m/s}^2$.

17.- A wheel with a diameter $20\ \text{cm}$ is rotating at $3000$ r.p.m. When a brake is applied to the wheel, we notice that it stops after $20\ \text{s}$. Find: (a) The angular acceleration (if we assume it to be constant) and the number of revolutions undergone by the wheel until it stops. (b) The tangential and normal accelerations of a point on the edge of the wheel once it has undergone one hundred revolutions, and the resultant acceleration at that point.

18.- A lighthouse rotates with a constant angular velocity $\omega$. If the lighthouse is placed at a distance $d$ from a completely straight beach, find: (a) The velocity and the linear acceleration that the light spot moves on the beach when the angle between $d$ and the light beam is $\theta$.

19.- A bullet is launched from the ground level with no air resistance, with an inclination of $40^\circ$ above the horizontal and an initial velocity of $200\ \text{m/s}$. (a) Find the velocity and the position of the bullet $20\ \text{s}$ after its launching. (b) How far from the launching point will the bullet reach the ground level again? Find the time elapsed until the bullet reaches the ground level again.

20.- From a plane inclined an angle $\alpha$ with respect to the horizontal direction, we throw a stone with an initial velocity $v_0$ perpendicularly to the plane. Find the distance, measured along the inclined plane, from the launching point to the point when the stone reaches the incline plane again.

21.- A $1.5\ \text{m}$ tall boy is located $15\ \text{m}$ away from a $5\ \text{m}$ high wall. He throws a stone to the wall with a velocity forming an angle of $45^\circ$ with respect to the horizontal. What is the minimum value of the velocity the boy must release the stone so that it passes over the wall?

22.- A $3\ \text{m}$ high elevator cab rises with a constant acceleration of $1\ \text{m/s}^2$. When the elevator reaches a certain point, a chandelier lamp is released from the cab ceiling. Calculate the time elapsed until the chandelier lamp reaches the ground of the cab.