v( t) g 2 v 0 sin θ ( ) ( ) g t ( ) = 0

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Julian Fox
5 years ago
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1 PROJECTILE MOTION Velocity We seek to explore the velocity of the projectile, includin its final value as it hits the round, or a taret above the round. The anle made by the velocity vector with the local horizontal is also of interest. We start with the kinematic relations and the resultant vector velocity manitude is v x ( t) v y ( t) t which can be written as ) v 0 sin( θ) t The anle α of the velocity vector, with respect to the local horizontal, positive counterclockwise, at any time, is, from consideration of the vector trianle v y ( t) α( t) atan v x ( t) atan Aain consider the value at the vertex: v( t) : t where the constant in the time-parentheses is the time to attain the maximum y (vertex), as is shown in another paper in this series. So at the vertex the time component is zero and the velocity is just the x velocity. Thus the y-component of the velocity is zero at the vertex; this can also be seen with v 0 sin( θ) t vertex atan tan θ α t vertex t v v 0 sin( θ) 0 sin θ 0 v y v 0 sin θ 0 v 0 cos( θ) α( t) : atan tan θ Thus the velocity vector is horizontal at the vertex, so all the motion is in the x-direction. As time increases past the vertex point, the anle becomes neative, so that the motion is, as we already know, toward the round. Note that up to now the initial heiht has not entered the equations. If we wish to plot the anle as a function of time we need to consider that the fliht has a finite duration- time does not increase without limit in the physical situation. We have, from another paper, the eneral time of fliht (TOF; time from launch until the projectile reaches the round): ) t () () T : y 0 v 0 sin( θ) Now we can plot the time-dependent anle. The initial heiht is zero, the initial velocity is 0 m/s at an anle of 30 derees. We see that the anle is positive, decreases throuh zero at the vertex (which, for a zero initial heiht, occurs at half the TOF), and then becomes neative until it reaches the same manitude it had at the start (more on this below). If the initial heiht is nonzero, this is not the case. (c) W. C. Evans 004

2 40 Velocity anle, derees Time, s We can also develop this anle of the velocity vector usin calculus. Startin with the parametric equations of motion, x( t) v 0 cos( θ) t y( t) y0 v 0 sin( θ) t t the (time-dependent) derivative will be the tanent to the trajectory; for parametric equations this is dy dx dy dt dx dt t tan θ v 0 cos( θ) t which is the arument of the arctanent in Eq(). Note that the manitude of the velocity was found in Eq(), usin the kinematic equations, while the anle of the velocity vector is found in cartesian coordinates, as we just did. Another way to show this is of course to use the cartesian form of the trajectory (developed in another paper in this series), namely: dy dx y( x) ) x x tan θ v 0 cos( θ) tan( θ) x y0 t ) tan θ ) which is the same as we just found, above, usin the parametric form. We can now write the velocity vector's anle in terms of x: α( x) atan x tan θ ) (3) (c) W. C. Evans 004

3 Next we develop the relations for the end of the fliht, just as the projectile hits the round. At this point, the time is the TOF, so that v x ( T) T v y ( T) y 0 v 0 sin( θ) v y ( T) ( ) y 0 (4) The manitude of the resultant velocity is v( T) v x v y ( ) y 0 ) so that the "final" velocity manitude will be v f v 0 y 0 (5) We can also derive this from conservation of enery: m v 0 m y 0 m v f m 0 It can be seen by inspection that this produces the same result. This function has several interestin properties. First, the final velocity does not depend on the initial anle. Second, when the initial heiht is zero, the initial and final velocities are equal in manitude. Third, as the initial velocity increases, the final velocity approaches the initial, reardless of the initial heiht. That is, lim v 0 v 0 y 0 v 0 A plot of Eq(5) will be of interest, but first, some analytic eometry. We can write Eq(5) as v f v 0 y 0 v f v 0 y 0 v f v 0 y 0 y 0 which is the form of a rectanular hyperbola: y a x b a b y 0 In this case the axis of symmetry is the y-axis, and we have a "rectanular" hyperbola since a b. The vertex is at "a" and it can be shown that the asymptotes are y a b x y a b x which here will be y x and y -x. Next we raph this function, with the initial velocity as x and the final velocity as y. (c) W. C. Evans 004 3

4 Final velocity, m/s Initial velocity, m/s The parameter in the plot is the initial heiht, at 0, 5, 0, 0, 30 m. A neative initial velocity corresponds to "throwin" the projectile downward. The dotted lines are the asymptotes, which are drawn by the zero initial heiht case, since then the initial and final velocities are equal. Here we have plotted the manitude of the final velocity, but the hyperbola has a neative branch as well, and we could just as well show the final velocity as neative (since it is directed downward). A zero initial velocity means that we just drop the object from its initial heiht. So, e.., from a heiht of 5 m, it takes about one second to reach the round, and the velocity will then be about 0 m/s, as the plot shows (y-intercept). A key fact, which is not intuitively obvious, is that an object projected with a iven initial velocity will hit the round with a particular final velocity (manitude), reardless of the initial anle. A special case of this is to throw a ball straiht up, from a bride, as opposed to throwin it straiht down from the bride, in both cases with the same initial velocity. The ball will hit the round with the same speed in either case. This is apparent in the symmetry of the hyperbolas in the fiure. Another example would be to throw the ball at an upward anle of, say, 60 derees, and then aain at a neative anle of 40 derees. The final velocity is the same in both cases. This is best seen from an enery viewpoint- the kinetic and potential eneries are the same no matter at what anle the ball is thrown. Also notice that the slope of the final-velocity curves is minimized for small values of the initial velocity. This can be shown with calculus, but it is apparent in the plot. The final velocity does not chane much from the free-fall value (the y-intercept) when the initial velocity is small. On the other hand, when the initial velocity is lare, the final velocity is essentially equal to it. (c) W. C. Evans 004 4

5 Another topic of interest is the anle of the velocity vector at impact. This would be α( T) atan tan θ T or α( R) atan tan θ R ) where R is the maximum x distance, or rane. The first one is a bit easier to work with, so we have α( T) atan tan θ y 0 v 0 sin( θ) ) which reduces to α( T) atan tan θ y 0 ) (6) From this we see that, if the initial heiht is zero, the final anle is the neative of the initial anle. Another interestin result is that, as the initial heiht becomes lare, we find lim atan tan θ y 0 y 0 ) π which says that the projectile hits the round travelin essentially straiht down. Similarly, as the initial velocity becomes lare, the final anle is aain the neative of the initial anle, reardless of the initial heiht or the initial anle's value. For completeness, we can write, usin the point-slope form, an equation for the tanent line at impact (recall that y is zero at impact): y tanent ( x) ( x R) tan θ y 0 ) (7) Finally, we can develop expressions for the velocity manitude and anle for the case of hittin a taret (at a nonzero heiht). The equations derived above will still be applicable, but the "TOF" will be different. For the taret case, the TOF is simple: T where is the x-coordinate of the taret. The anle θ needed to hit the taret is specifically calculated for some iven initial velocity and heiht (see the Taretin paper in this series). So, with this TOF, we have v x ( T) v y ( T) (c) W. C. Evans 004 5

6 and then the velocity manitude will be v( T) ) v 0 sin( θ) v 0 cos( θ) v 0 tan( θ) The velocity anle is found similarly: v α( T) atan tan( θ) 0 cos( θ) atantan θ v 0 cos( θ) ) (c) W. C. Evans 004 6

PROJECTILE MOTION. ( ) g y 0. Equations ( ) General time of flight (TOF) General range. Angle for maximum range ("optimum angle")

PROJECTILE MOTION. ( ) g y 0. Equations ( ) General time of flight (TOF) General range. Angle for maximum range (optimum angle) PROJECTILE MOTION Equations General time of fliht (TOF) T sin θ y 0 sin( θ) General rane R cos( θ) T R cos θ sin( θ) sin( θ) y 0 Anle for maximum rane ("optimum anle") θ opt atan y 0 atan v f atan v f