projectile motion

projectile motion overview

• a projectile is an object not in contact with the ground that is allowed to move under the influence of gravity and no other external force

  • acceleration of the object is constant - uniform acceleration

• human body can be considered a projectile whenever it is airborne

  • running, jumping, falling, etc.

• due to constant acceleration on everything due to gravity we can use equations to predict where an object that is in flight will be if we know certain variables

• an object in flight on earth will always follow a parabolic path

  • use calculus to derive the equations of motion and use these equations to solve for an objects kinetic values as it travels through space

parabolic flight path

• need to think about force vercotrs and derive equations of motion

• we know that when we throw an object it will follow a parabolic path

• when object is released or thrown it has only one force of gravity acting on it

• need to consider x and y components separately

• if object is thrown vertically acceleration equation would be a_y(t) = -9.81m/s²

  • to determine velocity and displacement we would need to integrate acceleration

projectile features

1. at apex vertical velocity is zero

2. apex is the higher point achieved

3. if a projectile takes off and lands at the same height then

   1. apex occurs at ½ the total flight time

   2. vertical component of velocity at the beginning of the flight is equal but opposite to the vertical component at end of flight

horizontal projective kinematics

• horizontal velocity stays constant throughout the flight

  • results in horizontal displacement defined by the time that it takes the object ot rise and fall vertically

  • both vertical and horizontal kinematics of a projectile that defines the parabolic flight path

• V_x(t) = V_0x

• to find the equation for displacement we find the integral

• d_x(t) = v_oxt + d_ox

horizontal and vertical velocity

• both components together demonstrate why the object has a parabolic path

• adding both components allows us to see resultant velocity vectors in the direction of the parabolic flight

  • as we would expect

optimal angles of release

• optimal angle for height is different than for distance

• assuming magnitude of velocity cannot be altered

  • optimal angle for height is 90º or straight up

  • optimal angle for distance is 45º

• looking at initial velocity to optimizer both time in air and horizontal displacement

vertical projectile motion equations

a_y(t) = -9.81 m/s²

V_y(t) = -9.81 m/s² + V_oy

d_y(t) = -9.81 m/s² + V_oy + d_oy

horinzontal projectile motion equations

a_y(t) = 0 m/s²

V_y(t) = V_ox

d_y(t) = V_ox + d_ox

common equipment for projectile motion

• jump mat

  • measures time in air and outputs jump height

  • used operational amplifiers with acceleration and velocity to calculate jump height

• force plate

  • measures time n the air and calculates initial velocity

• iPod (IMU integrated w/I phone)

  • jump time app

  • measures period of time when accelerometer doesn’t measure any force

    • time in air and equations to determine jump height

projectile motion equation strategies

1. determine what is asked for and draw diagram

2. list all known variables

   1. include those given in the question and those that can be solved without calculation

3. derive the equations off option and choose equation that allows you to use your variables to solve for the answer ot the question

   1. may take more than one step to solve for other unknowns first

4. plug in variables and solve for answer

5. check if variables seem reasonable