12 x1 t04 05 displacement, velocity, acceleration (2012)
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Applications of Calculus To The Physical World
Applications of Calculus To The Physical World
Displacement (x)Distance from a point, with direction.
Applications of Calculus To The Physical World
Displacement (x)Distance from a point, with direction.
x
dtdxv ,,Velocity
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.
x
dtdxv ,,Velocity
The rate of change of displacement with respect to time i.e. speed with direction.
vxdt
xddtdva ,,,,on Accelerati 2
2
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.
x
dtdxv ,,Velocity
The rate of change of displacement with respect to time i.e. speed with direction.
vxdt
xddtdva ,,,,on Accelerati 2
2
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.
Displacement
Velocity
Acceleration
differentiate
Displacement
Velocity
Acceleration
differentiate integrate
t
x
1 2 3 4
12
-2-1
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
x
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
(iii) average speed m/s47
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
(iii) average speed m/s47
(iv) average velocity m/s41
23 21ttx
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
a) Find the acceleration of the particle at time t.
23 21ttx
426423
212
23
tattv
ttx
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
a) Find the acceleration of the particle at time t.
23 21ttx
426423
212
23
tattv
ttx
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
a) Find the acceleration of the particle at time t.
b) Find the times when the particle is stationary.
23 21ttx
426423
212
23
tattv
ttx
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
a) Find the acceleration of the particle at time t.
0423i.e. 2 tt
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
23 21ttx
426423
212
23
tattv
ttx
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
a) Find the acceleration of the particle at time t.
0423i.e. 2 tt
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
14or 00143
tttt
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.
cttvta
232
62
(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.
cttvta
232
62
0,0when vt
0000 i.e.
cc
(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.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv
(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.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
(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.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
7,0when xt
7007 i.e.
cc
(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.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
7,0when xt
7007 i.e.
cc
732 ttx
(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.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
7,0when xt
7007 i.e.
cc
732 ttx
43 733,3when 32
xt
after 3 seconds the particle is 43 units to the right of O.
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
(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
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
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
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
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
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
22
tt
412
xt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
2
4 12
vt
242
vt
22
tt
412
xt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
2
4 12
vt
242
vt
1
4
4 2 2 12
ta
t
382
at
22
tt
412
xt
(iii) Is the particle ever at rest? Give reasons for your answer.
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
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.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
0
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
0
OR v
t
1 242
vt
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
0
OR v
t
1
the limiting velocity of the particle is 0 m/s
242
vt
(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 sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t
(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 sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
(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 sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(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 sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(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 sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(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 sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(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 sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
sin 2 3x t
(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
0dxdt
i.e. the stationary points
(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
when seconds, 4 metres4
t x
3 seconds, 2 metres4
t x
5 seconds, 4 metres4
t x
7 seconds, 2 metres4
t x
0dxdt
i.e. the stationary points
(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
when seconds, 4 metres4
t x
3 seconds, 2 metres4
t x
5 seconds, 4 metres4
t x
7 seconds, 2 metres4
t x
0dxdt
i.e. the stationary points
(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
when seconds, 4 metres4
t x
3 seconds, 2 metres4
t x
5 seconds, 4 metres4
t x
7 seconds, 2 metres4
t x
0dxdt
i.e. the stationary points
The particle oscillates between x=2 and x=4 with a period ofseconds
Exercise 3B; 2, 4, 6, 8, 10, 12