Category: 02 Kinematics

2-5-6 Trajectory Comparison

Same uy, different ux

classicProjectile1

  • All three projectiles have the same vertical motion since they have the same initial vertical velocity uy.
  • Since the time of flight is the same, the horizontal range is proportional to horizontal velocity ux.

Same ux, different uy

classicProjectile2.gif

  • The maximum height is proportional to uy2.
  • All three projectiles actually move forward at the same speed since they have the same horizontal velocity ux.
  • But the horizontal range is also dependent on uy, resulting in the horizontal range being also proportional ot uy.

Same θ, different u

classicProjectile3.gif

  • Both the maximum height and horizontal range are proportional to u2.

Same u, different θ

classicProjectile4.gif

  • Increasing θ results in increasing uy but decreasing ux.
  • The projectile launched at θ = 15° lands at the spot at the once launched at θ = 75°. This is because even though it moves forward at a higher speed, it stays in flight for a shorter time.
  • The maximum range is attained at θ = 45°.

2-5-5 Horizontal Projectile

This demonstration illustrates the fact that the vertical and horizontal motion of a projectile motion are independent of each other.

The coins were launched with the same initial vertical velocity of zero, but different horizontal velocities. Since the vertical motion is totally determined by the initial vertical velocity, all the coins drop vertically at the same rate, and land at the same time. The horizontal velocity determines how fast the coin moves forward. Since they land at the same time, the coin with the highest horizontal velocity lands furthest away.

2-4 Diluted Acceleration of Free Fall

The cute thing about uniform acceleration motion is that the acceleration is constant, the velocity changes linearly, and the displacement changes quadratically. So if the motion starts from rest, then the displacement at equal time intervals is going to show up as the sequence of square numbers: 0, 1, 4, 9, 16, 25, 36,…

Rolling Down an Incline

Bouncing Ball

2-3 Acceleration of Free Fall

Aristotle philosophized that heavier objects should fall faster than light objects. Galileo hypothesized that all objects fall at the same rate.

Galileo had only water clocks and his eyes to work with. But he brilliantly proofed his case by rolling balls down inclines. (The dropping of balls from the tower of Pisa is fake news)

Today, we have vacuums and high speed cameras.

Apollo 15

https://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html

Brian Cox

Brainiac

2.1.2 Kinematics Graphs

Worked Example

Given the v-t graph for the motion of a ball over 5 seconds, derive the corresponding s-t and a-t graphs. (Assume “rightward is positive” sign convention.)

example

Solution

kinegraph1

  • The areas of +1.0, +0.5, -0.5 and -1.0 in the v-t graph correspond to the s-t graph increasing by 1.0 m and 0.5 m, before decreasing by 0.5 m and 1.0 m.
  • As the ball continues to move rightward until t = 2 s, the maximum displacement occurs at t = 2 s.
  • The ball returns to the starting point after 5 s because the positive area happens to match the negative area exactly.

kinegraph2

  • As the velocity changes, so does the gradient of the v-t
  • Constant speed segments in v-t graphs translate into straight lines in s-t graphs.
  • Increasing speed segments translate into steepening curves, whereas decreasing speed segments translate into flattening curves.

kinegraph3

  • The a-t graph is readily obtained from the gradient of the v-t

kinegraph4

  • The areas of 0, -2.0 and +1.0 in the a-t graph correspond to velocity remaining constant, before decreasing by 2.0 m s-1 and then increasing by 1.0 m s-1.

2.1.1 Sign Convention

  • Let’s adopt the “rightward is positive” convention for the motions below

1 pospos

  • Since the sheep is moving rightward and speeding up, its velocity is represented by more and more positive numbers.
  • Since the velocity is becoming more positive, the acceleration is positive.

1 pospos

  • Since the sheep is moving leftward and speeding up, its velocity is represented by more and more negative numbers.
  • Since the velocity is becoming more negative, the acceleration is negative.

1 pospos

  • Since the sheep is moving rightward but slowing down,  its velocity is represented by less and less positive numbers.
  • Since the velocity is becoming less positive, the acceleration is negative.

1 pospos

  • Since the sheep is moving leftward but slowing down,  its velocity is represented by less and less negative numbers.
  • Since the velocity is becoming less negative, the acceleration is positive.
  • In a nutshell, the speed increases if the velocity and acceleration have the same sign, and decreases if the velocity and acceleration have opposite signs.

207 Quadratic s-t

The cute thing about uniform acceleration motion is that the acceleration is constant, the velocity changes linearly, and the displacement changes quadratically. So if the motion starts from rest, then the displacement at equal time intervals is going to show up as the sequence of square numbers: 0, 1, 4, 9, 16, 25, 36,…

Rolling Down an Incline

Bouncing Ball

205 Coin Projectiles

This demonstration illustrates the fact that the vertical and horizontal motion of a projectile motion are independent of each other.

The coins were launched with the same initial vertical velocity of zero, but different horizontal velocities. Since the vertical motion is totally determined by the initial vertical velocity, all the coins drop vertically at the same rate, and land at the same time. The horizontal velocity determines how fast the coin moves forward. Since they land at the same time, the coin with the highest horizontal velocity lands furthest away.

204 Ting Ting Ting

Ting, ting, ting…

Did you see/hear the ball ringing the vertical lines at constant time intervals? Does it ring a bell in you that the ball is moving forward at a constant speed?

Yup. The force of gravity is the only (significant) force acting on the ball when it was in the air. Since gravity is constant and acts vertically downward, it does not affect the horizontal motion.

539892_large

That makes projectile motion very easy to analyse. We calculate the vertical and horizontal motion separately. The vertical motion is uniform acceleration motion at g = 9.81 m s-2 (downward), whereas the horizontal motion is constant speed motion.