18. Exercises

As a car moves along a straight rod the distance, \displaystyle s metres, of a car from the origin at time \displaystyle t seconds is given by:

\displaystyle s=\frac{{{t}^{3}}}{3}-\frac{{{t}^{4}}}{60} for \displaystyle 0\le t\le 10.

a) By differentiating, find an expression for the velocity of the car at time \displaystyle t.

b) Find an expression for the acceleration of the car at time \displaystyle t.

c) Find the times when the acceleration of the car is zero.

A particle, of mass 4 kg, accelerates from rest so that the distance that it has travelled in \displaystyle t seconds is \displaystyle s where \displaystyle s=5{{t}^{2}}-6t.

a) Find the velocity and acceleration of the particle.

b) Find the time when the velocity is zero.

c) Find the magnitude of the resultant force on the particle.

A crane lifts a load from ground level. The height, \displaystyle s m, of the lift at time \displaystyle t seconds is given by \displaystyle s=\frac{3{{t}^{2}}}{50}-\frac{{{t}^{3}}}{250} for \displaystyle 0\le t\le 10.

a) Show that the velocity of the load is zero when \displaystyle t=\text{ 1}0.

b) Find the height of the load at this time.

c) Find the time when the acceleration of the load is zero.

d) Find the height of the load at this time.

The distance, \displaystyle s m, travelled by a dragster at time \displaystyle t seconds is given by,

\displaystyle s=18{{t}^{2}}-{{t}^{3}}

This expression only applies until the acceleration of the dragster becomes zero for the first time.

a) Find the time when the acceleration of the dragster is zero.

b) Find the speed of the dragster at this time.

c) Find the maximum acceleration of the dragster.

A particle moves so that its displacement, \displaystyle s m, at time \displaystyle t seconds is given by \displaystyle s=k{{t}^{2}}-\frac{5{{t}^{3}}}{3}.

a) Show that the particle is at rest when \displaystyle t~=\text{ }0.

b) Find k, if the particle comes to rest when \displaystyle t~=\text{ 2}0.

c) Sketch an acceleration-time graph for the particle.

d) Find the time when the acceleration of the particle is zero.

A particle, of mass 5 kg, moves so that its position vector, \displaystyle \mathbf{r} metres, at time \displaystyle t seconds is given by:

\displaystyle \mathbf{r}=(12-2{{t}^{2}})\mathbf{i}+(10-5{{t}^{2}})\mathbf{j}

The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are perpendicular.

a) Find the position of the particle when \displaystyle t=\text{ 1}0.

b) Find the speed of the particle when \displaystyle t=\text{ 2}.

c) Find the magnitude of the resultant force on the particle.

An aeroplane moves so that at time \displaystyle t seconds, its position vectors, \displaystyle \mathbf{r} metres, is given by:

\displaystyle \mathbf{r}=( 40t )\mathbf{i}+( 3{{t}^{2}}+20t )\text{ }\mathbf{j}

The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are that are directed horizontally and vertically respectively. The aeroplane is initially at ground level.

a) Find the time when the height of the aeroplane is 500 metres.

b) Find the speed of the aeroplane at this time.

c) Show that the acceleration of the aeroplane is constant and state its magnitude.

A boat moves so that its position vector, \displaystyle ~\mathbf{r} metres, at time \displaystyle t seconds is given by \displaystyle \mathbf{r}=\left( 2t-\frac{{{t}^{2}}}{10} \right)\mathbf{i}+2t\mathbf{j}

The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are directed east and north respectively.

(a) Find the time when the boat is due north of its initial position.

b) Find an expression for the speed of the boat at time \displaystyle t.

c) Find the time when the boat is travelling north.

d) Show that the acceleration of the boat is constant and state the direction of the acceleration.

The position vector, \displaystyle \mathbf{r} metres, of a particle at time \displaystyle t seconds is given by:

\displaystyle \mathbf{r}=({{t}^{2}}-t+5)\mathbf{i}+({{t}^{3}}-12{{t}^{2}}+45t+17)\mathbf{j}

The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are horizontal and vertical unit vectors respectively. The mass of the particle is 3 kg.

a) Find an expression for the velocity of the particle at time \displaystyle t.

b) Find the times when the particle is moving parallel to the ground.

c) Find an expression for the resultant force acting on the particle at time \displaystyle t.