Mechanics — Variant 2

A particle of mass $1.6$ kg is dropped from a height of $9$ m above horizontal ground. The speed of the particle at the instant before hitting the ground is $12$ $\text{m}\,\text{s}^{-1}$. Find the work done against air resistance.

Two particles $A$ and $B$, of masses $3.2$ kg and $2.4$ kg respectively, lie on a smooth horizontal table. $A$ moves towards $B$ with a speed of $v$ $\text{m}\,\text{s}^{-1}$ and collides with $B$, which is moving towards $A$ with a speed of $6$ $\text{m}\,\text{s}^{-1}$. In the collision the two particles come to rest.

Find the value of $v$.

Find the loss of kinetic energy of the system due to the collision.

An athlete of mass $84$ kg is running along a straight road.

Initially the road is horizontal and he runs at a constant speed of $3$ $\text{m}\,\text{s}^{-1}$. The athlete produces a constant power of $60$ W. Find the resistive force which acts on the athlete.

The athlete then runs up a $150$ m section of the road which is inclined at $0.8^\circ$ to the horizontal. The speed of the athlete at the start of this section of road is $3$ $\text{m}\,\text{s}^{-1}$ and he now produces a constant driving force of $24$ N. The total resistive force which acts on the athlete along this section of road has constant magnitude $13$ N. Use an energy method to find the speed of the athlete at the end of the $150$ m section of road.

A particle $P$ starts at rest and moves in a straight line from a point $O$. At time $t$ s after leaving $O$, the velocity of $P$, $v$ $\text{m}\,\text{s}^{-1}$, is given by $v = bt + ct^{\frac{3}{2}}$, where $b$ and $c$ are constants. $P$ has velocity $8$ $\text{m}\,\text{s}^{-1}$ when $t = 4$ and has velocity $13.5$ $\text{m}\,\text{s}^{-1}$ when $t = 9$.

Show that $b = 3$ and $c = -0.5$.

Find the acceleration of $P$ when $t = 1$.

Find the positive value of $t$ when $P$ is at instantaneous rest and find the distance of $P$ from $O$ at this instant.

Find the speed of $P$ at the instant it returns to $O$.