FALLING BODIES AND PROJECTILE MOTION

Galileo showed that two metal spheres of different mass would strike the ground at the same time when dropped from a building. He deduced that if it were not for air friction all objects would fall at the same rate. He was right. The development of vacuum pumps have shown this to be true. A feather and a metal ball fall at the same rate in an evacuated tube or on the moon.

Experiments show that the acceleration due to gravity (g) on earth averages about 9,8 m.s-2.

The actual value varies around the world due to:

1.the shape of the earth (g greater at poles).

2.the spin of the earth.

3.the variations in density around the world.

NOTE: if a body is falling or rising freely (i.e. only the force due to gravity acting on it) then it will accelerate at 9,8 m.s-2down.

It is VERY important in all calculations that you define a positive direction.

Unless otherwise stated, always assume that there is no air resistance for graphs and calculations.

When there is no air resistance a body projected up at velocity v will return to the same horizontal height at velocity -v In addition the time to go up will equal the time to come down.

MOTION

/ or
or / or

Example:How high will a ball rise above where it is released if it takes 4,3 s to reach to top of its flight after being thrown up?

Graphs for projectile motion

An object, travelling at some initial velocity, slides up a frictionless incline. It slows down uniformly, changes direction, then accelerates down the incline to the bottom again. Assuming that its acceleration is uniform throughout and that its velocity on return has the same magnitude as when it started, draw sketch graphs of:

a)displacement vs timed) speed vs time

b)distance vs timee) acceleration vs time

c)velocity vs time

Take up the slope as the positive direction.

GRAPHS OF MOTION

DISPLACEMENT VS TIME

  1. If velocity is constant then the graph has straight lines. (Δx α t)
  1. If the body is accelerating then the lines will be curved (Δx α t2)
  1. The gradient is the velocity of the body.

VELOCITY VS TIME

  1. Only constant acceleration is asked and therefore the lines are always straight (i.e. not curved).
  1. The area under the curve is the displacement.
  1. The gradient is the acceleration.

ACCELERATION VS TIME

  1. The gradient of these will always be zero because we only work with constant acceleration.
  1. Remember a body slowing down is actually accelerating in the opposite direction (i.e. it has negative acceleration).
  1. The area under the line is the change in velocity of the body.

SOME TIPS

  1. Always determine whether you are dealing with a vector quantity.
  1. Define a direction as positive (usually the initial direction) and stick to it.
  1. Ask yourself if the body’s motion is constant velocity or constant acceleration.
  1. Remember a body dropped or rising freely is accelerating at the same rate (9,8 m.s-2 down) if friction with the air is ignored.
  1. Label your axes.
  1. Check your answer, following the curve and seeing that the change on the y-axis follows the motion of the body.

Bouncing balls

A ball bouncing is accelerating down at 9,8 m.s-2 all the time except when in contact with the ground. If we assume that the time in contact with the ground is very small then the velocity-time graph for a ball being dropped from a height and bouncing 3 times before being caught is:

velocity

+

time

-

Note:

  1. Up is positive in the above graph.
  2. The gradient is negative and its value is -9,8 m.s-2 every time.
  3. Kinetic energy is converted to heat and sound every time the ball bounces and so its initial velocity after bouncing drops each time.
  4. At the highest point each time the velocity is zero.

The position-time graph is:

position

time

Terminal velocity (extension but also Newton II so could be asked)

If a body falls it initially accelerates at “g” but as it moves air opposes this motion. The body will therefore experience 2 forces, weight down (W) and air resistance up (F). The faster the body moves the greater will be the air resistance until a speed is reached when the magnitude of the air resistance is equal to the weight of the body and then the resultant force will be zero. With no resultant force there will be no acceleration and terminal velocity will have been reached.

A steel ball was dropped from a height of about 4000 m. Taking air resistance into account, state which of the following are true and which are false.

a)Its speed increases steadily with time.

b)Its speed increases steadily with distance.

c)Its acceleration stays constant.

d)Its speed increases to a maximum and then stays constant.

e)Its acceleration increases to a maximum and then stays constant.

f)Its acceleration increases to a maximum and then decreases.

g)Its speed increases to a maximum and then decreases to a constant smaller value.

h)Its acceleration decreases to zero as it falls.

i)The maximum resultant force exerted on the body is when it is about to hit the ground.

j)The maximum resultant force exerted on the body is when it has just started falling.

k)It has a maximum acceleration when it is about to hit the ground.

l)It has a maximum acceleration when it is has just started falling.

m)The retarding force due to air friction increases to a maximum when it is equal

to the mass of the ball.

Projectile motion 2-D (additional science)

An object projected through a gravitational field is known as a projectile.

When an object is projected at an angle to the horizontal its flight will take a curved path. Its flight can be regarded as the combination of two flights – vertical under the influence of gravity and horizontal (at constant velocity). (We always assume no air resistance in calculations.)

The initial vertical (vy) and horizontal (vx) velocities can be calculated because they are components of the initial velocity (v) (previous notes).

At position Q the object only has a horizontal velocity (still of vx.)

The total horizontal distance it travels is called the RANGE.

Examples:

1.A ball is projected at 34 m.s-1 at 40o to the horizontal over flat ground.

Calculate:a) how long it is in the air? (Advice: do not use Δx = vi.t + ½a.Δt2)

b) how far the ball travels horizontally?

2.A cannon fires a shell from a cliff 60 m above the ground at a ship X km from the base of the cliff and hits it. If the shell left the muzzle at a speed of 320 m.s-1 and was projected horizontally, calculate the value of X.