The document discusses key concepts in calculus that are applied to physical systems, including: - Displacement (distance from a point with direction) - Velocity (rate of change of displacement over time, including direction) - Acceleration (rate of change of velocity over time) It provides examples of how to calculate acceleration, velocity, and displacement from equations describing physical phenomena. The relationships between these concepts are illustrated, such as how displacement can be determined by integrating velocity or differentiating acceleration. Calculus allows describing physical systems quantitatively and analyzing properties like when an object is stationary.

1. Applications of Calculus To
The Physical World

2. Applications of Calculus To
The Physical World
Displacement (x)
Distance from a point, with direction.

3. Applications of Calculus To
The Physical World
Displacement (x)
Distance from a point, with direction.
 v, dx , x 
Velocity  
 dt 
The rate of change of displacement with respect to time i.e. speed
with direction.

4. Applications of Calculus To
The Physical World
Displacement (x)
Distance from a point, with direction.
 v, dx , x 
Velocity  
 dt 
The rate of change of displacement with respect to time i.e. speed
with direction.
 dv d 2 x 
Acceleration  a, , 2 , , v 
x 
 dt dt 
The rate of change of velocity with respect to time

5. Applications of Calculus To
The Physical World
Displacement (x)
Distance from a point, with direction.
 v, dx , x 
Velocity  
 dt 
The rate of change of displacement with respect to time i.e. speed
with direction.
 dv d 2 x 
Acceleration  a, , 2 , , v 
x 
 dt dt 
The rate of change of velocity with respect to time
NOTE: “deceleration” or slowing down is when acceleration is in
the opposite direction to velocity.

6. Displacement
differentiate Velocity
Acceleration

7. Displacement
differentiate Velocity integrate
Acceleration

8. x
2
1
1 2 3 4 t
-1
-2

9. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1
-2

10. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1
-2

x
1 2 3 4 t

11. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1
-2
slope=instantaneous acceleration

x
1 2 3 4 t

12. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1
-2
slope=instantaneous acceleration

x
1 2 3 4 t

x
1 2 3 4 t

13. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1 e.g. (i) distance traveled  7 m
-2
slope=instantaneous acceleration

x
1 2 3 4 t

x
1 2 3 4 t

14. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1 e.g. (i) distance traveled  7 m
-2
slope=instantaneous acceleration (ii) total displacement  1m

x
1 2 3 4 t

x
1 2 3 4 t

15. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1 e.g. (i) distance traveled  7 m
-2
slope=instantaneous acceleration (ii) total displacement  1m

x
7
(iii) average speed  m/s
4
1 2 3 4 t

x
1 2 3 4 t

16. x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1 e.g. (i) distance traveled  7 m
-2
slope=instantaneous acceleration (ii) total displacement  1m

x
7
(iii) average speed  m/s
4
1
1 2 3 4 t (iv) average velocity  m/s
4

x
1 2 3 4 t

17. e.g. (i) The displacement x from the origin at time t seconds, of a
particle traveling in a straight line is given by the formula
x  t 3  21t 2
a) Find the acceleration of the particle at time t.

18. e.g. (i) The displacement x from the origin at time t seconds, of a
particle traveling in a straight line is given by the formula
x  t 3  21t 2
a) Find the acceleration of the particle at time t.
x  t 3  21t 2
v  3t 2  42t
a  6t  42

19. e.g. (i) The displacement x from the origin at time t seconds, of a
particle traveling in a straight line is given by the formula
x  t 3  21t 2
a) Find the acceleration of the particle at time t.
x  t 3  21t 2
v  3t 2  42t
a  6t  42
b) Find the times when the particle is stationary.

20. e.g. (i) The displacement x from the origin at time t seconds, of a
particle traveling in a straight line is given by the formula
x  t 3  21t 2
a) Find the acceleration of the particle at time t.
x  t 3  21t 2
v  3t 2  42t
a  6t  42
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
i.e. 3t 2  42t  0

21. e.g. (i) The displacement x from the origin at time t seconds, of a
particle traveling in a straight line is given by the formula
x  t 3  21t 2
a) Find the acceleration of the particle at time t.
x  t 3  21t 2
v  3t 2  42t
a  6t  42
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
i.e. 3t 2  42t  0
3t t  14   0
t  0 or t  14
Particle is stationary initially and again after 14 seconds

22. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.

23. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.
a  2  6t
v  2t  3t 2  c

24. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.
a  2  6t
v  2t  3t 2  c
when t  0, v  0
i.e. 0  0  0  c
c0

25. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.
a  2  6t
v  2t  3t 2  c
when t  0, v  0
i.e. 0  0  0  c
c0
 v  2t  3t 2

26. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.
a  2  6t
v  2t  3t 2  c
when t  0, v  0
i.e. 0  0  0  c
c0
 v  2t  3t 2
x  t2  t3  c

27. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.
a  2  6t
v  2t  3t 2  c
when t  0, v  0
i.e. 0  0  0  c
c0
 v  2t  3t 2
x  t2  t3  c
when t  0, x  7
i.e. 7  0  0  c
c7

28. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.
a  2  6t
v  2t  3t 2  c
when t  0, v  0
i.e. 0  0  0  c
c0
 v  2t  3t 2
x  t2  t3  c
when t  0, x  7
i.e. 7  0  0  c
c7
 x  t2  t3  7

29. (ii) A particle is moving on the x axis. It started from rest at t = 0 from
the point x = 7.
If its acceleration at time t is 2 + 6t find the position of the particle
when t = 3.
a  2  6t when t  3, x  32  33  7
v  2t  3t 2  c  43
when t  0, v  0
after 3 seconds the particle is 43
i.e. 0  0  0  c
units to the right of O.
c0
 v  2t  3t 2
x  t2  t3  c
when t  0, x  7
i.e. 7  0  0  c
c7
 x  t2  t3  7

30. e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?

31. e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t  0, x 
02
= 1
 the particle is 1 metre to the left of the origin

32. e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t  0, x 
02
= 1
 the particle is 1 metre to the left of the origin
4
(ii) Show that x  1 
t2
Hence find expressions for the velocity and the acceleration in terms of t.

33. e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t  0, x 
02
= 1
 the particle is 1 metre to the left of the origin
4
(ii) Show that x  1 
t2
Hence find expressions for the velocity and the acceleration in terms of t.
4 t 24
1 
t2 t2
t2
 4
t  2  x  1
t2

34. e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t  0, x 
02
= 1
 the particle is 1 metre to the left of the origin
4
(ii) Show that x  1 
t2
Hence find expressions for the velocity and the acceleration in terms of t.
4 t 24 4  1
1  v
t2 t2 t  2
2
t2
 4 v
4
t  2  x  1  t  2
2
t2

35. e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t  0, x 
02
= 1
 the particle is 1 metre to the left of the origin
4
(ii) Show that x  1 
t2
Hence find expressions for the velocity and the acceleration in terms of t.
t 24 4  1 4  2  t  2  1
1
4
1  v a
t2 t2 t  2  t  2
2 4
t2 8
 4 v
4
a
t  2  x  1  t  2
2
 t  2
3
t2

36. (iii) Is the particle ever at rest? Give reasons for your answer.

37. (iii) Is the particle ever at rest? Give reasons for your answer.
4
v 2  0
t  2
 the particle is never at rest

38. (iii) Is the particle ever at rest? Give reasons for your answer.
4
v 2  0
t  2
 the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?

39. (iii) Is the particle ever at rest? Give reasons for your answer.
4
v 2  0
t  2
 the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?
4
lim v  lim
t 
 
t  t  2 2

40. (iii) Is the particle ever at rest? Give reasons for your answer.
4
v 2  0
t  2
 the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?
4
lim v  lim
t 
 
t  t  2 2
0

41. (iii) Is the particle ever at rest? Give reasons for your answer.
4
v 2  0
t  2
 the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?
4 v
lim v  lim OR
t 
 
t  t  2 2
4
v
0 1 t  2
2
t

42. (iii) Is the particle ever at rest? Give reasons for your answer.
4
v 2  0
t  2
 the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?
4 v
lim v  lim OR
t 
 
t  t  2 2
4
v
0 1 t  2
2
t
 the limiting velocity of the particle is 0 m/s

43. (ii) 2002 HSC Question 8b)
A particle moves in a straight line. At time t seconds, its distance x
metres from a fixed point O in the line is given by x  sin 2t  3
(i) Sketch the graph of x as a function of t for 0  t  2

44. (ii) 2002 HSC Question 8b)
A particle moves in a straight line. At time t seconds, its distance x
metres from a fixed point O in the line is given by x  sin 2t  3
(i) Sketch the graph of x as a function of t for 0  t  2
2 
amplitude  1 unit period  divisions 
2 4
shift  3 units 

45. (ii) 2002 HSC Question 8b)
A particle moves in a straight line. At time t seconds, its distance x
metres from a fixed point O in the line is given by x  sin 2t  3
(i) Sketch the graph of x as a function of t for 0  t  2
2 
amplitude  1 unit period  divisions 
2 4
shift  3 units 
x
4
3
2
1
  3  5 3 7 2 t
4 2 4 4 2 4

46. (ii) 2002 HSC Question 8b)
A particle moves in a straight line. At time t seconds, its distance x
metres from a fixed point O in the line is given by x  sin 2t  3
(i) Sketch the graph of x as a function of t for 0  t  2
2 
amplitude  1 unit period  divisions 
2 4
shift  3 units 
x
4
3
2
1
  3  5 3 7 2 t
4 2 4 4 2 4

47. (ii) 2002 HSC Question 8b)
A particle moves in a straight line. At time t seconds, its distance x
metres from a fixed point O in the line is given by x  sin 2t  3
(i) Sketch the graph of x as a function of t for 0  t  2
2 
amplitude  1 unit period  divisions 
2 4
shift  3 units 
x
4
3
2
1
  3  5 3 7 2 t
4 2 4 4 2 4

48. (ii) 2002 HSC Question 8b)
A particle moves in a straight line. At time t seconds, its distance x
metres from a fixed point O in the line is given by x  sin 2t  3
(i) Sketch the graph of x as a function of t for 0  t  2
2 
amplitude  1 unit period  divisions 
2 4
shift  3 units 
x
4
3
2
1
  3  5 3 7 2 t
4 2 4 4 2 4

49. (ii) 2002 HSC Question 8b)
A particle moves in a straight line. At time t seconds, its distance x
metres from a fixed point O in the line is given by x  sin 2t  3
(i) Sketch the graph of x as a function of t for 0  t  2
2 
amplitude  1 unit period  divisions 
2 4
shift  3 units 
x
4 x  sin 2t  3
3
2
1
  3  5 3 7 2 t
4 2 4 4 2 4

50. (ii) Using your graph, or otherwise, find the times when the particle is at
rest, and the position of the particle at those times.

51. (ii) Using your graph, or otherwise, find the times when the particle is at
rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
dx
 0 i.e. the stationary points
dt

52. (ii) Using your graph, or otherwise, find the times when the particle is at
rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
dx
 0 i.e. the stationary points
dt

when t  seconds, x  4 metres
4
3
t seconds, x  2 metres
4
5
t seconds, x  4 metres
4
7
t seconds, x  2 metres
4

53. (ii) Using your graph, or otherwise, find the times when the particle is at
rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
dx
 0 i.e. the stationary points
dt

when t  seconds, x  4 metres
4
3
t seconds, x  2 metres
4
5
t seconds, x  4 metres
4
7
t seconds, x  2 metres
4
(iii) Describe the motion completely.

54. (ii) Using your graph, or otherwise, find the times when the particle is at
rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
dx
 0 i.e. the stationary points
dt

when t  seconds, x  4 metres
4
3
t seconds, x  2 metres
4
5
t seconds, x  4 metres
4
7
t seconds, x  2 metres
4
(iii) Describe the motion completely.
The particle oscillates between x=2 and x=4 with a period of
 seconds

55. Exercise 3B; 2, 4, 6, 8, 10, 12