Linear Motion

3.1

Motion is Relative:

• All motion is relative.
• If unstated, the motion mentioned is relative to the surface of the Earth.
• Even things that are at rest are moving.
• You’re moving at 170,000 km/h right now.

3.2

Speed:

• Before Galileo, people described things as either slow or fast.
• Galileo was the first to measure speed.
• Speed = Distance/Time
• Galileo measured distance easily. He struggled with measuring time.
  • Sometimes he used his own pulse, or dripping water drops.
• Eg: Cyclist covering 16 meters in 2 seconds has a speed of 8 meters per second (m/s).

Instantaneous Speed:

• Instantaneous speed is the speed of an object at any instant.
  • If a car travelled at 50 km/h for an hour, it would cover 50 km. If it travelled for half an hour, it would cover 25 km.

Average Speed:

• Average speed = total distance covered/time interval
• Eg: if you travel 320 km in 4 hrs, the average speed is 80 km/h.
• Average speed doesn’t tell you the different speeds at smaller time intervals.
• Average speed is different from instantaneous speed.
• Total distance covered = average speed x time interval

3.3

Velocity:

• The speed and direction of motion for an object gives us its velocity.
  • Speed → car travelling at 60 km/h.
  • Velocity → car moving at 60 km/h towards north.
• Speed is how fast an object moves.
• Velocity is how fast an object moves and in what direction it moves.
• Vector quantity: Quantity that specifies magnitude and direction.
• Scalar quantity: Quantity that specifies only magnitude.

Constant Velocity:

• Constant speed is steady speed.
• Constant velocity is constant speed and constant direction.
• Constant direction means the object moves in a straight line. The path doesn’t curve.
• Constant velocity → Motion in a straight line at a constant speed.

Changing Velocity:

• Velocity changes if speed or direction or both change.

3.4

Acceleration:

• Velocity of an object changes with a change in speed, change in direction, or change in both.
• Acceleration → How quickly the velocity changes and in what direction.
• Formula → Acceleration = change of velocity/time interval
• Acceleration is defined by change.
  • Eg: increasing velocity from 30 km/h to 35 km/h in one second, to 40 km/h in the next, 45 km/h in the next, and so on.
  • Here, the acceleration is 5 km/h.s
• Acceleration is the change per second in the velocity.
• Acceleration is the increase or decrease in the velocity.
• Deceleration → When there is a large decrease per second in the velocity.
• When we move in a curved path, our direction is constantly changing, so we are accelerating, even if we are moving at a constant speed.
  • Eg: Standing in a moving bus. When a bus moves at a constant velocity, you can stand with no extra effort. When the bus accelerates, you experience difficulty standing.

Acceleration on Galileo’s Inclined Planes:

• Galileo demonstrated acceleration using experiments on inclined planes.
• A ball rolling down an inclined plane picks up the same speed with each successive second. This is constant acceleration.
  • Eg: For a ball rolling down a plane inclined at an angle, let’s say it picks up a speed of 2 m/s with each second.
  • The instantaneous velocity at 1s intervals at this acceleration is 0, 2, 4, 6, 8, 10 m/s, and so on.
  • Velocity acquired = acceleration x time
  • At the end of 1s, the ball travels 2 m/s, at the end of 2s, it is travelling 4 m/s, at the end of 10s, it’s 20 m/s.
• Acceleration down an incline is constant for each incline.
• Steeper inclines have greater accelerations.
  • When the incline is tipped vertically, the ball accelerates to its highest extent. It falls with the acceleration of a falling object.
• When air resistance can be ignored, all objects fall with the same acceleration.

3.5

Free Fall:

How Fast:

• When an object falls under the influence of only gravity, it is in a state of free fall.
  • There’s no other restraints like friction with the air acting on the object.
• Instantaneous velocity of a free falling object at 1 second intervals:

• During each second, the object gains a speed of 10 m/s.
• Gain per second is the acceleration.
• Free fall acceleration = 10 m/s^2.
• For free falling objects, g represents acceleration.
  • Acceleration is due to gravity.
• g is slightly different on the surface of the Moon and the surfaces of other planets.
• Average value of g = 9.8 m/s2
• Instantaneous velocity, v = gt
• Speedometers can be used to measure free fall acceleration.
• For an object thrown upwards:
  • It slows down as it moves upwards.
  • At its highest point, it changes direction.
  • At its highest point, its instantaneous speed becomes 0.
  • When it starts moving downwards, it acts like its been dropped from rest at that height.
  • Upward deceleration = Downward acceleration
  • The velocities act the opposite ways, because they’re acting in opposite directions.
  • Downward velocity has a negative sign, indicating downward direction.

How Far:

• Inclined planes showed Galileo that the distance of a uniformly accelerating object is proportional to the square of time taken.
• Distance travelled = 1/2 x acceleration x time x time
• Shorthand notation: d = 1/2gt^2

• d → distance object falls
• t → time taken for the fall

Note:

→ For an object falling 5m during the first second of a 10m/s fall, it’d be expected to cover 10m. This can only actually happen if the average speed of the object is 10m/s for that entire second.

→ The average speed in this second is actually the sum of the starting and final speeds, divided by 2.

→ (0+10)/2, which is 5m/s.

• Objects fall with unequal accelerations.
  • Eg: a leaf, a feather, and a sheet of paper fall at different speeds due to different air resistances.
  • This can be demonstrated with a closed glass tube containing these objects.
  • If the air in the tube is replaced with a vacuum, the objects will fall at the same speed.
• Heavier objects aren’t appreciably affected by air resistance.
• v = gt and d = 1/2 gt^2 tell us about objects falling in the air from an initial state of rest.

How Quickly ‘How Fast’ Changes:

• For an object falling, we are talking about speed or velocity.
  • v = gt
• For how far an object falls, we are talking about distance.
  • d = 1/2 gt^2
• Velocity is a rate (rate of change of position).
• Acceleration is a rate of a rate (rate of change of velocity).

Hang Time:

• Athletes and dancers appear to “hang in the air” for 2 to 3 seconds when they jump.
• This is actually one 1 second.
• People can easily cross over a 0.5 m gate, but they wouldn’t be able to jump 0.5 m high.
• It’s easier for people to leap over a fence than to shift their center of gravity entirely.
• When you jump upwards, you only apply force when your feet are in contact with the ground. As soon as you’re in the air, your upward speed decreases at a rate of 10m/s^2. At the topmost point, your speed is 0.
• When you start to fall, your speed increases at the same rate.
• Hang rate is the sum of the rising time and the falling time.
• Relationship between up or down and vertical height:
  • d = 1/2gt^2 , or,
  • t = square root of (2*d/g)

3.6

Velocity Vectors:

• Speed → how fast

• Velocity → how fast + in what direction

• Speed is a scalar quantity. Velocity is a vector quantity.

  • Eg: Consider an airplane flying 80 km/h due north getting caught in a 60 km/h crosswind and getting thrown off course by it.

  A crosswind is a wind blowing at a 90˚angle to the object being mentioned. Here, the plane.

  • If the 80 km/h vector is 4 cm long, and the 60 km/h vector is 3 cm, the resultant of the two vectors is a 100km/h vector measuring 5 cm long, at an angle of 37˚ from the 60 km/h vector.
  • 
  • The plane is due eastward.