18. Exercises
From Mechanics
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b) Find the time when the velocity is zero. | b) Find the time when the velocity is zero. | ||
| - | c) Find the magnitude of the resultant force on the particle | + | c) Find the magnitude of the resultant force on the particle. |
</div>{{#NAVCONTENT:Answer a|Answer 18.2a|Answer b|Answer 18.2b|Answer c|Answer 18.2c|Solution a|Solution 18.2a|Solution b|Solution 18.2b|Solution c|Solution 18.2c}} | </div>{{#NAVCONTENT:Answer a|Answer 18.2a|Answer b|Answer 18.2b|Answer c|Answer 18.2c|Solution a|Solution 18.2a|Solution b|Solution 18.2b|Solution c|Solution 18.2c}} | ||
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seconds is given by, | seconds is given by, | ||
| - | <math>s= | + | <math>s=18{{t}^{2}}-{{t}^{3}}</math> |
This expression only applies until the acceleration of the dragster becomes zero for the first time. | This expression only applies until the acceleration of the dragster becomes zero for the first time. | ||
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</div>{{#NAVCONTENT:Answer a|Answer 18.4a|Answer b|Answer 18.4b|Answer c|Answer 18.4c|Solution a|Solution 18.4a|Solution b|Solution 18.4b|Solution c|Solution 18.4c}} | </div>{{#NAVCONTENT:Answer a|Answer 18.4a|Answer b|Answer 18.4b|Answer c|Answer 18.4c|Solution a|Solution 18.4a|Solution b|Solution 18.4b|Solution c|Solution 18.4c}} | ||
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===Exercise 18.5=== | ===Exercise 18.5=== | ||
<div class="ovning"> | <div class="ovning"> | ||
| + | A particle moves so that its displacement, | ||
| + | <math>s</math> | ||
| + | m, at time | ||
| + | <math>t</math> | ||
| + | seconds is given by | ||
| + | <math>s=k{{t}^{2}}-\frac{5{{t}^{3}}}{3}</math>. | ||
| + | a) Show that the particle is at rest when | ||
| + | <math>t~=\text{ }0</math>. | ||
| + | |||
| + | b) Find k, if the particle comes to rest when | ||
| + | <math>t~=\text{ 2}0</math>. | ||
| + | |||
| + | c) Sketch an acceleration-time graph for the particle. | ||
| + | |||
| + | d) Find the time when the acceleration of the particle is zero. | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | </div>{{#NAVCONTENT:Answer b|Answer 18.5b|Answer c|Answer 18.5c|Answer d|Answer 18.5d|Solution a|Solution 18.5a|Solution b|Solution 18.5b|Hint c|Hint 18.5c|Solution d|Solution 18.5d}} | ||
| - | </div>{{#NAVCONTENT:Answer a|Answer 18.5a|Answer b|Answer 18.5b|Answer c|Answer 18.5c|Solution a|Solution 18.5a|Solution b|Solution 18.5b|Solution c|Solution 18.6c}} | ||
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===Exercise 18.6=== | ===Exercise 18.6=== | ||
<div class="ovning"> | <div class="ovning"> | ||
| + | |||
| + | A particle, of mass 5 kg, moves so that its position vector, | ||
| + | <math>\mathbf{r}</math> | ||
| + | metres, at time | ||
| + | <math>t</math> | ||
| + | seconds is given by: | ||
| + | |||
| + | <math>\mathbf{r}=(12-2{{t}^{2}})\mathbf{i}+(10-5{{t}^{2}})\mathbf{j}</math> | ||
| + | |||
| + | The unit vectors <math>\mathbf{i}</math> and <math>\mathbf{j}</math> are perpendicular. | ||
| + | |||
| + | a) Find the position of the particle when | ||
| + | <math>t=\text{ 1}0</math>. | ||
| + | |||
| + | b) Find the speed of the particle when | ||
| + | <math>t=\text{ 2}</math>. | ||
| + | |||
| + | c) Find the magnitude of the resultant force on the particle. | ||
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<div class="ovning"> | <div class="ovning"> | ||
| + | An aeroplane moves so that at time | ||
| + | <math>t</math> | ||
| + | seconds, its position vectors, | ||
| + | <math>\mathbf{r}</math> | ||
| + | metres, is given by: | ||
| + | |||
| + | <math>\mathbf{r}=( 40t )\mathbf{i}+( 3{{t}^{2}}+20t )\text{ }\mathbf{j}</math> | ||
| + | |||
| + | The unit vectors <math>\mathbf{i}</math> and <math>\mathbf{j}</math> 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. | ||
</div>{{#NAVCONTENT:Answer a|Answer 18.7a|Answer b|Answer 18.7b|Answer c|Answer 18.7c|Solution a|Solution 18.7a|Solution b|Solution 18.7b|Solution c|Solution 18.7c}} | </div>{{#NAVCONTENT:Answer a|Answer 18.7a|Answer b|Answer 18.7b|Answer c|Answer 18.7c|Solution a|Solution 18.7a|Solution b|Solution 18.7b|Solution c|Solution 18.7c}} | ||
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| + | |||
| + | ===Exercise 18.8=== | ||
| + | <div class="ovning"> | ||
| + | |||
| + | A boat moves so that its position vector, | ||
| + | <math>~\mathbf{r}</math> | ||
| + | metres, at time | ||
| + | <math>t</math> | ||
| + | seconds is given by | ||
| + | <math>\mathbf{r}=\left( 2t-\frac{{{t}^{2}}}{10} \right)\mathbf{i}+2t\mathbf{j}</math> | ||
| + | |||
| + | The unit vectors <math>\mathbf{i}</math> and <math>\mathbf{j}</math> 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 | ||
| + | <math>t</math>. | ||
| + | |||
| + | 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. | ||
| + | |||
| + | </div>{{#NAVCONTENT:Answer a|Answer 18.8a|Answer b|Answer 18.8b|Answer c|Answer 18.8c|Answer d|Answer 18.8d|Solution a|Solution 18.8a|Solution b|Solution 18.8b|Solution c|Solution 18.8c|Solution d|Solution 18.8d}} | ||
| + | |||
| + | |||
| + | ===Exercise 18.9=== | ||
| + | <div class="ovning"> | ||
| + | |||
| + | The position vector, | ||
| + | <math>\mathbf{r}</math> | ||
| + | metres, of a particle at time | ||
| + | <math>t</math> | ||
| + | seconds is given by: | ||
| + | |||
| + | <math>\mathbf{r}=({{t}^{2}}-t+5)\mathbf{i}+({{t}^{3}}-12{{t}^{2}}+45t+17)\mathbf{j}</math> | ||
| + | |||
| + | The unit vectors <math>\mathbf{i}</math> and <math>\mathbf{j}</math> 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 | ||
| + | <math>t</math>. | ||
| + | |||
| + | 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 | ||
| + | <math>t</math>. | ||
| + | |||
| + | |||
| + | </div>{{#NAVCONTENT:Answer a|Answer 18.9a|Answer b|Answer 18.9b|Answer c|Answer 18.9c|Solution a|Solution 18.9a|Solution b|Solution 18.9b|Solution c|Solution 18.9c}} | ||
Current revision
| Theory | Exercises |
Exercise 18.1
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.
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Exercise 18.2
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.
Exercise 18.3
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.
Exercise 18.4
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.
Exercise 18.5
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.
Exercise 18.6
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.
Exercise 18.7
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.
Exercise 18.8
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.
Exercise 18.9
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.
