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.
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
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.
x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1
-2
x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1
-2
x
1 2 3 4 t
x
2 slope=instantaneous velocity
1
1 2 3 4 t
-1
-2
slope=instantaneous acceleration
x
1 2 3 4 t
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
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
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
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
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
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.
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
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.
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
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
(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.
(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
(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
c0
(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
c0
v 2t 3t 2
(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
c0
v 2t 3t 2
x t2 t3 c
(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
c0
v 2t 3t 2
x t2 t3 c
when t 0, x 7
i.e. 7 0 0 c
c7
(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
c0
v 2t 3t 2
x t2 t3 c
when t 0, x 7
i.e. 7 0 0 c
c7
x t2 t3 7
(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.
c0
v 2t 3t 2
x t2 t3 c
when t 0, x 7
i.e. 7 0 0 c
c7
x t2 t3 7
e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t 0, x
02
= 1
the particle is 1 metre to the left of the origin
e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t 0, x
02
= 1
the particle is 1 metre to the left of the origin
4
(ii) Show that x 1
t2
Hence find expressions for the velocity and the acceleration in terms of t.
e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t 0, x
02
= 1
the particle is 1 metre to the left of the origin
4
(ii) Show that x 1
t2
Hence find expressions for the velocity and the acceleration in terms of t.
4 t 24
1
t2 t2
t2
4
t 2 x 1
t2
e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t 0, x
02
= 1
the particle is 1 metre to the left of the origin
4
(ii) Show that x 1
t2
Hence find expressions for the velocity and the acceleration in terms of t.
4 t 24 4 1
1 v
t2 t2 t 2
2
t2
4 v
4
t 2 x 1 t 2
2
t2
e.g. 2001 HSC Question 7c)
A particle moves in a straight line so that its displacement, in metres,
is given by t2
x
t2
where t is measured in seconds.
(i) What is the displacement when t = 0?
02
when t 0, x
02
= 1
the particle is 1 metre to the left of the origin
4
(ii) Show that x 1
t2
Hence find expressions for the velocity and the acceleration in terms of t.
t 24 4 1 4 2 t 2 1
1
4
1 v a
t2 t2 t 2 t 2
2 4
t2 8
4 v
4
a
t 2 x 1 t 2
2
t 2
3
t2
(iii) Is the particle ever at rest? Give reasons for your answer.
(iii) Is the particle ever at rest? Give reasons for your answer.
4
v 2 0
t 2
the particle is never at rest
(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?
(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
(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
(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
(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
(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
(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
(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
(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
(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
(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
(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
(ii) Using your graph, or otherwise, find the times when the particle is at
rest, and the position of the particle at those times.
(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
(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
(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.
(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