clickandlearn.org

Projectile motion is the motion of an object launched or propelled at an angle q with respect to the horizontal

The standard equations describing the motion of an object in two dimensions can be applied to this special case of accelerated motion (under the influence of the acceleration due to gravity).

a) Important points to realize when analyzing projectile motion:

• The projectile angle is q
• The initial velocity V₁ is tangent to the path of motion at an angle q with respect to the horizontal therefore, it has horizontal and verticalcomponents.
• The horizontal component of the initial velocity is called V₁x; where V₁x = V₁ (cos q)
  (see vector addition for review of this topic)
• The vertical component of the initial velocity is called V₁y; where V₁y = V₁ (sin q)
• The trajectory (the path of the projectile) is a parabola. The parabola is symmetrical about a point half way between the origin and the total horizontal distance traveled by the projectile.
• At the half time point (t_1/2), the projectile reaches its maximum height.
• The total horizontal distance (from the origin to where the projectile lands) is called the range (R).
• When the projectile lands at time t (the total flight time), its vertical distance is zero (ground level)
• A projectile path (because it is a parabolic trajectory) will have two roots
• Projectiles can be analyzed as a function [y = f(t)] of time t, and as a function of horizontal
  distance x [y = (fx)]

b) Important Physics principles of projectiles:

1. The horizontal velocity V_x, is always constant -- not affected by gravity (i.e. a = 0)/
2. The vertical velocity V_y, is affected by gravity -- it is positive but decreasing for the first part of the trajectory and negative but increasing for the second part of the trajectory. (i.e. a = -9.8 m/s²).
3. At the top of its flight (at t_1/2), the slope of the tangent to this parabola is zero, therefore the vertical velocity V_y at that point is zero.
4. Considering points 1, and 2 above, a projectile launched at angle q and an object dropped at the same time from the same height, will reach the ground at the same time. See illustration below...

c) Equations of motion relating to projectiles -- and solution tips!

Projectile Motion equations and Linear Motion Equations
Linear Equations of Motion	Projectile Motion
	Horizontal	Vertical
v2 = v1 + a Dt		V2y = V1y+ 2g Dt
v22 = v12 + 2 a Dd		V2y2 = V1y2 + 2g y
Dd = v1Dt + 1/2 at2 Dd = Vav ct (use in non-accelerated motion)	The horizontal distance is calculated by: x = V1 Dt The Range (R) is the maximum horizontal distance: R = V12 (sin 2q )/g This is known as the Range Equation	The Height (h) can be calculated by: y = Vy + 1/2g(Dt2) To find the maximum height substitute Vy = 0; at any other time use Dt to find the height of the projectile
vav = (v1 + v2) / 2	To find the total time use y = 0	Maximum Height can be also be found by using Dt = t1/2
Use in free fall to calculate height: Dd(y)= v1Dt + 1/2 gDt2	V1x = V1 (cos q) This gives you the initial horizontal velocity	V1y = V1 (sin q) This gives you the initial vertical velocity
	This gives you the instantaneous velocity at any point along the trajectory of the projectile Use Pythagoras' Theorem to find the magnitude of Vtot. Use trigonometry to find its direction.
Substitute a with g (where g = - 9.8 m/s2) when dealing with free fall or projectiles.