Projectile motion, is motion with constant acceleration in one fixed direction. Rewriting the well known displacement equation [which describes motion in a straight line with constant acceleration] as a vector equation gives....
Writing this equation with column vectors makes calculations easier.
The vector equation defines the trajectory (path) of the projectile. To find the position of the mass (x, y) at any time, substitute ux, uy, (the initial velocity in the x direction and y directions), and the time t in the equation.
Imagine that a ball is thrown straight up. It comes to rest at the top and then accelerates downwards to hit the ground at the same speed it started with (in the other direction).
Imagine now that the same ball is thrown upwards with the same vertical speed but this time it has a constant horizontal velocity. The motions of the two balls are the same. The only difference is the constant horizontal displacement of the second ball.
| At once: if any two balls reach the same height they spend the same total time in the air! Note also: the equation is in t squared - the path is a parabola. |
To show experimentally that the motion of a projectile follows a parabolic path it is convenient to photograph the stream of water from a hose and fit a known parabola to the curve. The fitting process is equally successful with a reduced launch angle.
The two equations which describe projectile motion are....
The Range is given by....
... where t is the time of flight.
Taking just the bottom line of the vector equation shows that the time of flight is given by....
Taking just the top line of the vector equation shows that the Range is uxt and the range is given by.....
When the initial conditions are given in terms of the speed of the projectile, u, and the angle of launch angle, q , above the horizontal, ux = u sinq and uy = u sinq. The range formula becomes ...
For a given initial kinetic energy, the maximum range occurs when cosq sinq is a maximum.
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Inspecting the graph above (made in graphing calculator) shows that the range is a maximum when the angle of launch is 45° above the horizontal.
The maximum height is given by the value of y when t is half the time of flight, which gives....
The trajectory equation holds only in the absence of air resistance.
 An ammunition dump explosion in Fallujah. Note the effect of air resistance: the trajectories are not symmetrical. Note also the rising rotating ring-vortex that forms the head of a mushroom cloud. |
During the initial explosive phase of the Arenal eruption (Costa Rica: 1968), projectiles were thrown up to 5 km. Considering the effect of atmospheric drag, these projectiles must have had initial velocities of at least 600 m/s. For projectiles to reach this velocity, the maximum gas pressure in the magma chamber must have been at least 4700 bars. Had the effect of aerodynamic braking been ignored in making these calculations, (as has always been done in the past), the calculated initial velocity would have been 220 m/s and chamber pressure estimates would have been substantially lower.
Edited extract from Fudali and Melson:1970,
Department of Mineral Sciences,
The Smithsonian Institution,
20560 Washington, D. C.
Note: the 1968 eruption ejected blocks of magma and caused hot avalanches. The western flank of the mountain was destroyed, and 78 people died. Arenal has erupted many times in recorded history, and very frequently for the last forty years.
Holes are drilled in a bottle. The pressure at a particular hole, is proportional to the head of water H. The time of flight is given at once by....
... where ux is constant and D is the range.
The height of the hole above the table h is given by....
Eliminating t gives at once....
It would be interesting to plot ux against the depth of water H. It would also be interesting to find within what limits the velocity of efflux is independent of hole diameter for holes at the same height.
Projectile motion occurs whenever a constant unbalanced force (in size and direction) is applied to an isolated mass. If an air table is inclined at a very small angle the path of the puck becomes parabolic. To a first approximation, neglecting the effects of rotation, a large steel ball rolling on an inclined table has the same motion.
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