2 | Motion
IB Physics Content Guide
• Motion is described relative to a chosen coordinate system.
• Vector quantities can be combined to find resultant vectors or divided into their component parts
• Displacement-time, velocity-time, and accel-time graphs are connected in the representation of physical motion.
• When an object is at constant velocity, displacement-time is linear.
• When an object is at constant acceleration, displacement-time is quadratic (curved), and velocity-time is linear.
• Kinematic equations can take three of the suvat variables to solve for the remaining two
• X and Y motion are independent of each other for a two-dimensional projectile
2 | Motion
How far (m)
How fast (m s-1)
Displacement vs Time
Velocity vs Time
Acceleration vs Time
Meaning of the Graph
Area under the Curve:
Area under the Curve:
Constant Positive Velocity
Constant Negative Velocity
Constant Positive Acceleration
Constant Negative Acceleration
0 m s-1
0 m s-2
-9.81 m s-2
0 m s-1
0 m s-2
-9.81 m s-2
MOTION IN A STRAIGHT LINE
DISTANCE AND DISPLACEMENT
- Displacement may be positive, negative or zero but distance is always positive.
- Displacement is not affected by the shift of the coordinate axes.
- Displacement of an object is independent of the path followed by the object but distance depends upon path.
- Displacement and distance both have same unit as that of length i.e. metre.
- For a moving body distance always increases with time
- For a body undergoing one dimensional motion, in the same direction distance = | displacement |. For all other motion distance > | displacement |.
- | Average velocity | can be zero but average speed cannot be zero for a moving object.
- | Instantaneous velocity | = Instantaneous speed.
- A particle may have constant speed but variable velocity. It happens when particle travels in curvilinear path.
- If the body covers first half distance with speed v1 and next half with speed v2 then
- If a body covers first one-third distance at a speed v1, next one-third at speed v2 and last one-third at speed v3, then
- If a body travels with uniform speed v1 for time t1 and with uniform speed v2 for time t2, then
EQUATIONS FOR UNIFORMLY ACCELERATED MOTION
- v = u + at
- v2 – u2 = 2as
v = final velocity
s = distance travelled in time t
- Distance travelled in nth second
sn = ;
VERTICAL MOTION UNDER GRAVITY
- For a body thrown downward with initial velocity u from a height h, the equations of motion are
- If initial velocity is zero, then the equations are
- When a body is thrown upwards with initial velocity u, the equations of motion are
UNIFORMLY ACCELERATED MOTION : A DISCUSSION
- The direction of average acceleration vector is the direction of the change in velocity vector
- There is no definite relationship between velocity vector and acceleration vector.
- For a body starting from rest and moving with uniform acceleration, the ratio of distances covered in t1 sec.,
t2 sec, t3 sec, etc. are in the ratio t12 : t22 : t32 etc.
- A body moving with a velocity v is stopped by application of brakes after covering a distance s. If the same body moves with a velocity nv, it stops after covering a distance n2s by the application of same retardation.
- An object moving under the influence of earth’s gravity in which air resistance and small changes in g are neglected is called a freely falling body.
- In the absence of air resistance, the velocity of projection is equal to the velocity with which the body strikes the ground.
- Distance travelled by a freely falling body in 1st second is always half of the numerical value of g or 4.9 m, irrespective of height h.
- For a freely falling body with initial velocity zero
- Velocity ∝ time (v = gt)
- Velocity (v2 = 2gs)
- Distance fallen α (time)2 , where g is the acceleration due to gravity.
- If maximum height attained by a body projected vertically upwards is equal to the magnitude of velocity of projection, then velocity of projection is 2g ms–1 and time of flight is 4 sec.
- If maximum height attained by a body projected upward is equal to magnitude of acceleration due to gravity i.e., ‘g’, the time of ascent is sec. and velocity of projection is .
- Ratio of maximum heights reached by different bodies projected with velocities u1, u2, u3 etc. are in the ratio of etc. and ratio of times of ascent are in ratio of u1 : u2 : u3 etc.
- During free fall velocity increases by equal amount every descend and distance covered during 1st, 2nd, 3rd seconds of fall, are 4.9m, 14.7m, 24.5m.
- If a body is projected horizontally from top of a tower, the time taken by it to reach the ground does not depend on the velocity of projection, but depends on the height of tower and is equal to .
- If velocity v of a body changes its direction by θ without change in magnitude then the change in velocity will be .
- From the top of a tower a body is projected upward with a certain speed, 2nd body is thrown downward with same speed and 3rd is let to fall freely from same point then
t2 = time taken by the body thrown downward and
t3 = time taken by the body falling freely.
- If a body falls freely from a height h on a sandy surface and it buries into sand upto a depth of x, then the retardation produced by sand is given by .
- In case of air resistance, the time of ascent is less than time of descent of a body projected vertically upward i.e. ta < td.
- When atmosphere is effective, then buoyancy force always acts in upward direction whether body is moving in upward or downward direction and it depends on volume of the body. The viscous drag force acts against the motion.
- If bodies have same volume but different densities, the buoyant force remains the same.
- When an aeroplane flying horizontally drops a bomb.
- An ascending helicopter dropping a food packet.
- A stone dropped from a moving train etc.
VARIOUS GRAPHS RELATED TO MOTION
- For a stationary body (v = 0) the time-displacement graph is a straight line parallel to time axis.
- When the velocity of a body is constant then time-displacement graph will be an oblique straight line. Greater the slope of the straight line, higher will be the velocity.
- If the velocity of a body is not constant then the time-displacement curve is a zig-zag curve.
- For an accelerated motion the slope of time-displacement curve increases with time while for decelerated motion it decreases with time.
- When the particle returns towards the point of reference then the time-displacement line makes an angle θ > 90° with the time axis.
- When the velocity of the particle is constant or acceleration is zero.
- When the particle is moving with a constant acceleration and its initial velocity is zero.
- When the particle is moving with constant retardation.
- When the particle moves with non-uniform acceleration and its initial velocity is zero.
- When the acceleration decreases and increases.
- The total area enclosed by the time – velocity curve represents the distance travelled by a body.
- When the acceleration of the particle is zero.
- When acceleration is constant
- When acceleration is increasing and is positive.
- When acceleration is decreasing and is negative
- When initial acceleration is zero and rate of change of acceleration is non-uniform
- The change in velocity of the particle = area enclosed by the time-acceleration curve.
RELATIVE VELOCITY (IN ONE DIMENSION)
2.1.1 Define displacement, velocity, speed and acceleration.
Displacement Displacement is the distance moved in a particular direction. It is a vector quantity.
SI unit: m Symbol: s
Velocity Velocity is the rate of change of displacement. It is a vector quantity. Velocity = (change in displacement / change in time)
SI unit: m s-1 Symbol: v or u
Speed Speed is the rate of change of distance. It is a scalar quantity. Speed = (change in distance / change in time)
SI unit: m s-1 Symbol: v or u
Note that speed and velocity are not the same thing. Velocity has a direction.
Acceleration Acceleration is the rate of change of velocity. It is a vector quantity. Acceleration = (change in velocity / change in time)
SI unit: m s-2 Symbol: a
Note that acceleration is any change in velocity, meaning an increase or decrease in velocity or a change in direction.
2.1.2 Explain the difference between instantaneous and average values of speed, velocity and acceleration.
Instantaneous An instantaneous value of speed, velocity or acceleration is one that is at a particular point in time.
Average An average value of speed, velocity or acceleration is one that is taken over a period of time.
2.1.3 Outline the conditions under which the equations for uniformly accelerated motion may be applied.
The equations of uniformly accelerated motion can only be under conditions where the acceleration is constant.
The equations of uniformly accelerated motion are as follows:
Table 1.2.1 – Variables used in uniformly accelerated motion equations
Other equations may be derived from these equations.
2.1.4 Identify the acceleration of a body falling in a vacuum near the Earth?s surface with the acceleration g of free fall.
When we ignore the effect of air resistance on an object falling down to earth due to gravity we say the object is in free fall. Free fall is an example of uniformly accelerated motion as the only force acting on the object is that of gravity.
On the earths surface, the acceleration of an object in free fall is about 9.81 ms-1. We can easily recognise the uniform acceleration in displacement – time, velocity – time and acceleration – time graphs as shown below:
2.1.5 Solve problems involving the equations of uniformly accelerated motion.
A car accelerates with uniformly from rest. After 10s it has travelled 200 m.
Its average acceleration
S = ut + 1/2 at²
200 = 0 x 10 + 1/2 x a x 10²
200 = 50a
a = 4 m s-2
Its instantaneous speed after 10s
v² = u ² + 2as
= 0 + 2 x 4 x 10
V= 8.9 m s-1
2.1.6 Describe the effects of air resistance on falling objects.
Air resistance eventually affects all objects that are in motion. Due to the effect of air resistance objects can reach terminal velocity. This is a point by which the velocity remains constant and acceleration is zero.
In the absence of air resistance all objects have the same acceleration irrespective of its mass.
2.1.7 Draw and analyse distance?time graphs, displacement?time graphs, velocity?time graphs and acceleration?time graphs.
2.1.8 Calculate and interpret the gradients of displacement?time graphs and velocity?time graphs, and the areas under velocity?time graphs and acceleration?time graphs.
Determining its velocity We know that the gradient of a displacement – time graph gives us its velocity. Therefore for the first 5 seconds the speed is:
After the first 5 s the object is stationary for 3 s. For these 3s its velocity is zero.
After 8s the object starts to return at a faster speed then before. From the graph we find the speed to be:
Figure 2.1.5 – Velocity -Time graph
Determine its acceleration We know that the gradient of a velocity- Time graph gives us its acceleration. Therefore for the first 5 s the acceleration is:
50/5 =10 ms?²
When the object is at constant speed from 5s to 7s its acceleration is zero. During the last second of the objects journey the object is decelerating at:
50/1 =50 ms?²
Determine its displacement The area under a velocity-time graph is the displacement. During the first 5 s the object has travelled:
½ x 5 x 50 = 125m
Determine the change in velocity The area under the acceleration- Time graph gives us the change in velocity
From the graph we find that the change in velocity is 10 x 3 = 30 ms?¹
Note: The gradient of the acceleration – time graph is actually the rate of change of acceleration. However it isn’t often useful.
|Displacement||Distance moved in particular direction||Vector|
|Velocity||Rate of change of displacement||Vector|
|Acceleration||Rate of change of velocity||Vector|
|Speed||Rate of change of distance||Scalar|
- Instantaneous speed, velocity or acceleration are quantities at certain points in time (ie. 2 seconds).
- Average quantities include all values at all points within a certain timeframe (ie. from 0 to 10 seconds).