Mathematics Extension 2

A particle is moving along a straight line. At time $t$, its velocity is $v$ and its displacement from a fixed origin is $x$.

If

which of the following best describes the particle’s acceleration and velocity?

(A) Constant acceleration and constant velocity

(B) Constant acceleration and decreasing velocity

(C) Constant acceleration and increasing velocity

(D) Increasing acceleration and increasing velocity

Using the same inclined-plane model, given that the object is initially at rest, find its velocity $\mathbf v(t)$ in terms of $g$, $\theta$, $t$ and $\mathbf i$.

An object of mass $m$ slides down a smooth inclined plane. The plane makes an angle $\theta$ with the horizontal. The normal reaction is $\mathbf R$, gravity is $\mathbf g$ with magnitude $g$, and $\mathbf i,\mathbf j$ are unit vectors parallel and perpendicular to the plane. Show that the resultant force is $\mathbf F=-(mg\sin\theta)\mathbf i.$

The acceleration of a particle is given by $\ddot{x}=\sec^2x$, where $x$ is the displacement in metres. The rule for the velocity could be:

A. $v=\sqrt{\tan x}$

B. $v=\sqrt{2\tan x}$

C. $v=\tan x$

D. $v=2\tan x$

A particle is moving in a straight line with acceleration $a=12-6t$. The particle starts from rest at the origin.

What is the position of the particle when it reaches its maximum velocity?

Hence, or otherwise, show that the point $P$ cannot be lifted to a position $\frac{2h}{3}$ metres above the floor.

A particle is travelling from $A$ on the curve joining $A$ to $B$. At a particular time, the particle is at $P$ and has velocity $\vec v$, as shown. The speed is increasing. Which diagram shows an acceleration $\vec a$ that would allow the particle to follow the curve to $B$?

There is one value of $v$ such that the object slides down the slope at constant speed. Find this value of $v$ in $\mathrm{m\,s^{-1}}$, correct to 1 decimal place, given that $g=10$.

A particle starts at the origin with velocity $1$ and acceleration given by
$a=v^2+v,$
where $v$ is the velocity of the particle. Find an expression for $x$, the displacement of the particle, in terms of $v$.

Find the velocity of the box after the first three seconds.

Show that the net force horizontally is approximately $53.2$ newtons.

A 50-kilogram box is initially at rest. The box is pulled along the ground with a force of $200$ newtons at an angle of $30^\circ$ to the horizontal. The box experiences a resistive force of $0.3R$ newtons, where $R$ is the normal force, as shown in the diagram. Take $g=10\,\mathrm{m/s^2}$.

(i) By resolving the forces vertically, show that $R=400$.