TOPIC : motion in a straight line
CLASS : XI
SUBJECT : PHYSICS
PREPARED BY:
M.G.ADHAU,
PGT-PHYSICS.
J.N.V.NANDED (M.S.)
To locate the position of object we require a reference point (origin) and a set of axes.
Describing motion
The coordinates (x,y,z) describe the position of object.
The coordinate system along with the clock is known as the frame of reference.
The actual path length travelled by the object is known as distance.
Distance:
It is Scalar quantity.
Displacement:
The change in position is known as displacement.
Displacement = Final position –Initial position
Δx = x2 – x1
x2 > x1, Δx– is positive
x2 < x1, Δx– is negative
The motion of the object can be represented by position –time graph.
Position –time Graph:(x-t graph)
t (S)
x (m)
0
10
20
30
40
10
20
30
40
t (S)
X (m)
0
1
2
3
4
2
4
6
8
Object is at rest
Object in uniform motion
Time (S)
Position(m)
0
1
2
3
4
2
4
6
8
Object in non-uniform motion
i.e. accelerated motion
Average velocity is defined as displacement (Δx) divided by time interval (Δt)
Average velocity:
It is measured in ms-1
Average Speed:
Average speed is defined as the total path length travelled by the object divided by time interval.
It is measured in ms-1
t (S)
X (m)
0
Q
A
B
P
X-t graph for two children A and B returning from their school O to their houses P and Q resp.
Choose the correct entries in the bracket
1.(A/B) lives closer to the school than(B/A)
2.(A/B) stars from the school earlier than(B/A)
3.(A/B) walks faster than(B/A)
4. A and B reach home at the (same/different) time.
5.(A/B) overtakes(B/A) on the road (once/twice)
Instantaneous Velocity:
The velocity at an instant is defined as the limit of average velocity as the time interval becomes infinitesimally small(Δt tends to zero)
The instantaneous velocity is also defined as the rate of change of position at that instant.
The instantaneous velocity is the time derivative of position.
Instantaneous Speed:
The magnitude of an instantaneous velocity is called as instantaneous speed.
Speed associated with both the velocities will 25 ms-1
Average acceleration is defined as change in velocity (Δv) divided by time interval (Δt)
Average Acceleration:
It is measured in ms-2
Instantaneous acceleration:
The acceleration at an instant is defined as the limit of average acceleration as the time interval becomes infinitesimally small (Δt tends to zero)
The instantaneous acceleration is also defined as the rate of change of velocity at that instant.
The instantaneous acceleration is the time derivative of velocity.
x
x
t
t
x
t
O
O
O
The Position –time graph
Positive
Acceleration
Zero Acceleration
Negative
Acceleration
t
O
The velocity time graph
Object is moving in positive direction with positive acceleration
v0
v
t
O
v0
v
Object is moving in positive direction with negative acceleration
t
O
Object is moving in negative direction with negative acceleration
-v0
-v
Object is moving with negative acceleration through out. Between 0 to t1 it moves + x-direction and between t1 to t2 it moves in opposite direction.
t2
O
v0
-v
t1
Time (h)
Velocity(km)/h
0
1
2
3
4
10
20
30
40
o
B
c
A
Area under Velocity-time Graph.
Velocity -Time Graph for object moving with constant velocity.
Area under
Velocity-Time graph
=
Area of Rectangle OABC
v
t
= OA x OC
= v x t
Product of velocity and time is equal to displacement.
Area under Velocity –Time graph is always equal to the distance travelled by an object.
t
0
E
B
C
t
v-v0
v0
v
D
A
Kinematic equations of uniformly accelerated motion
If the object is moving with constant acceleration then inst,acceleration is equal to av.acceleration
This is velocity-time relation
1.velocity-time relation
v = v0 +at
t
0
E
B
C
t
v-v0
v0
v
D
A
2. Position -time relation
Area under v-t graph.
A = Ar.of Rec.OACD +Ar of Tri. ACB
A = (AO x OD)+ ½ (Ac x BC)
A = (v0 x t)+ ½ (v-v0) x t
A = v0t + ½ (v0 +at-v0) x t
A = v0t + ½ a t2
x = v0t + ½ a t2
x –x0= v0t + ½ a t2
t
0
E
B
C
t
v-v0
v0
v
D
A
3. Position –velocity relation
Area under v-t graph.
Area of trapezium OABD
A = ½ x OD (OA +BD)
A = ½ x t (v0 +v)
A = ½ x (v-v0)/a x (v0 +v)
A = (v-v0)x (v + v0)/2a
A = (v2-v02)/2a
x = (v2-v02)/2a
x-x0 = (v2-v02)/2a
2a(x-x0 )= v2-v02
Home work
Equations of motion by Calculus method
1. velocity-time relation.
Acceleration
Integrating on both sides
This is velocity-time relation.
velocity
2. Position-time relation.
Integrating on both sides
This is Position-time relation.
3.Velocity-Position relation.
Integrating on both sides
This is velocity-Position relation.
Consider two objects A and B moving
uniformly with average velocities vA and vB along x-axis. If xA (0) and xB (0) are positions of objects A and B,respectively at time t = 0, their positions xA (t) and xB (t) at time t are given by:
xA (t ) = xA (0) + vA t --------1
xB (t) = xB (0) + vB t --------2
displacement of object B w.r.t. object A is
xBA(t) = xB (t) – xA (t)
xBA(t) = [ xB (0) – xA (0) ] + (vB – vA) t. ----3
Object B has a velocity vB – vA w.r.t. A
Relative velocity
The velocity of object B relative to object A is vB – vA
vBA = vB – vA
The velocity of object A relative to object B is vA – vB
vAB = vA – vB
O
20
40
60
80
100
120
140
X(m)
t(s)
O
20
40
60
80
100
120
140
t(s)
X(m)
1
2
3
4
5
6
1
2
3
4
5
6
A
B
A
B
vA = vB
vA – vB = 0
vB - vA = 0
The two objects stay at a constant distance
vA > vB
vB - vA is negative
Objects meet at a
common point
O
20
40
60
80
100
120
140
t(s)
X(m)
1
2
3
4
5
6
A
B
vA and vB are of opposite signs.
In this case, the magnitude of vBA or vAB is
greater than the magnitude of velocity of A or
that of B