DISCUSSION Projectile motion is one of the traditional branches of classical mechanics, with applications to ballistics. A projectile is any body that is given an initial velocity and then follows a path determined by the effect of the gravitational acceleration and by air resistance. Projectile motion is the motion of such a projectile. The free-flight motion of a projectile is often studied in terms of its rectangular components. To illustrate the kinematic analysis, consider a projectile launched at point (Xo, Yo), with an initial velocity of Vo, having components (Vo)x and (Vo)y.
In the figure, when air resistance is neglected, the only force acting on the projectile is its weight, which causes the projectile to have a constant downward direction of approximately a = g = 9.81m/s² or g = 32.2ft/s².
HORIZONTAL MOTION. Since ax = 0, application of the constant acceleration equations, yields +→ v = vo + act ; vx = (vo)x +→ x = xo + vot + ½ act² ; x = xo + (vo)xt +→ v² = vo² + 2ac (x-xo) ; vx = (vo)x The first and last equations indicate that the horizontal component of velocity always remains constant during the motion.
VERTICAL MOTION. Since the positive y-axis is directed upward, then ay = -g. Applying the equations, we get +↑ v = vo + act ; vy = (vo)y - gt +↑ y = yo + vot + ½ act² ; y = yo + (vo)yt + ½ gt² +↑ v² = vo² + 2ac (y-yo) ; vy² = (vo)y² - 2g(y-yo) Recall that the last equation can be formulated on the basis of eliminating the time t from the first two equations, and therefore only two of the above three equations are independent of one another.
Problem No. 1 A daredevil tries to jump a canyon of width 10 m. To do so, he drives his motorcycle up an incline sloped at an angle of 15 degrees. What minimum speed is necessary to clear the canyon? (Sample Problems) Solution>>>
Solution to Problem No. 1 First, write the constant velocity equation for the horizontal direction: vx = Dx / Dt = V cos15o Where V represents the total velocity at the 15o angle. Solving for V, V = Dx / (Dt cos15o ) Equation 1 Now write the constant acceleration equation for the vertical direction: Dy = (1/2) a Dt2 + vy Dt where vy = V sin15o substituting for vy: Dy = (1/2) a Dt2 + V sin15o Dt Equation 2 Now combine equations 1 and 2 by substituting for V: Dy = (1/2) a Dt2 + Dx / (Dt cos15 ) sin15o Dt simplify Dy = (1/2) a Dt2 + Dx sin15o / cos15o
Solve for Dt: Dt = [ 2( Dy - Dx sin15o / cos15o ) / a ]1/2 Substitute values: Dt = [ 2( 0m - 10m sin15o / cos15o ) / 10 m/s2 ]1/2 Dt = 0.73s Now insert the time into equation 1 to find V. V = 10m / (.73s) * cos15o) V = 14.18 m/s, the minimum speed needed for success. (Sample Problems)
Problem No. 2 An example of a projectile motion is shown in the figure, which shows a car rolling off a cliff. Suppose the care leaves the cliff with a velocity of 10m/s directed along the horizontal. If the cliff has a height h=20 m where and when will the car land? (Sample Problems) Solution>>>
Solution to Problem No. 2 y = h - ½gt2 0 = 20m - ½ (9.81m/s2)t2 t = 2.02secs x = Voxt x = 10m/s(2.02secs) x = 20.20meters (Sample Problems)
Problem No. 3 A projectile shot at an angle of 60o above the horizontal strikes a building 80 ft away at a point 48 ft above the point of projection. (a) Find the initial velocity, (b) Find the magnitude & direction of the velocity when it strikes the building. (Sample Problems) Solution>>>
Solution to Problem No. 3 The wording identifies the problem as a projectile motion problem. We draw a figure, choose a CS, and write down the initial conditions & acceleration in the problem. As before, since this is a 2-dimensional problem, initial position, initial velocity, and acceleration are specified by two numbers: xo = 0 vox = vo cos 60 = (.5) vo ax = 0 . yo = 0 voy = vo sin 60 = (.866) vo ay = - 32 ft/s2 . We note that the quantity vo is not given in the problem. Hence, our first task will be to determine this quantity. Inserting these values into the general equations of motion in 2-dimensions, we have: x(t) = (.5) vo t y(t) = - (1/2)(32) t2 + (.866)vo t vy(t) = - 32 t + (.866)vo . Since vo is not given in the problem, some other piece of information must be given. We read that the projectile: "strikes a building 80 ft away at a point 48 ft above the point of projection". Drawing a figure, we let the instant when the projectile strikes the building be: t. Then we have:
x(t) = 80 = (.5) vo t‘ y(t) = 48 = -(1/2)(32)t2 + (.866)vo t . Hence we have 2 equations in 2 unknowns and can solve for both t and vo. Solving we find: t = 2.38 seconds; vo = 67.3 ft/sec. Since we now know vo, then our specific equations of motion are complete, and we can calculate any other quantity associate with the motion. We are specifically asked for the velocity when it strikes the building (at time t = 2.38 sec.). Thus: vx = vox = (.5)(67.3) = 33.6 ft/sec vy(t=2.38s) = - 32(2.38) + (.866)(67.3) = - 17.9 ft/sec . We then draw the velocity vector from its components calculated above. The magnitude & direction (angle) can then be determined: v= = 38.1 ft/sec. tan = (17.9)/(33.6) = 28o (below hor. as shown) (Sample Problems)
Problem No. 4 A bullet has a speed of 350 m/sec as it leaves a rifle. If it is fired horizontally from a cliff 6.4 m above a lake, how far does the bullet travel before striking the water?(Sample Problems) Solution>>>
Solution to Problem No. 4 We have a 2-dimensional problem with constant acceleration (acceleration due to gravity). This is a projectile motion problem. The figure is as shown and the coordinate system selected is drawn. The origin is placed at the bullets location at time t=0. Hence the initial conditions for the problem are: x(t=0) = xo = 0 ; y(t=0) = yo = 0 vx(t=0) = vox = 350m/s ; vy(t=0) = voy = 0 Since the only force acting is gravity (downward = + y direction), we have: ax = 0; ay = + g = + 9.8 m/sec2. The general solutions for the constant acceleration problem in two dimensions are: x(t) = (1/2) ax t2 + vox t + xo y(t) = (1/2) ay t2 + voy t + yo vx(t) = ax t + vox vy(t) = ay t + voy
Inserting the values of acceleration and the initial conditions gives us the specific equations (applicable to this one particular problem). x(t) = (350) t y(t) = (1/2)(9.8) t2 vx = 350 m/s vy(t) = 9.8 t Let t be the time when the bullet hits the lake. We then know that: y(t) = + 6.4 m. Thus: y(t) = + 6.4 = + 4.9 t2 t = 1.143 sec. The horizontal (x) position of the bullet at this time is then: x(t) = (350)(1.143) = 400 m. (Sample Problem)
Problem No. 5 A player kicks a football at an angle of 37o with the horizontal and with an initial speed of 48 ft/sec. A second player standing at a distance of 100 ft from the first in the direction of the kick starts running to meet the ball at the instant it is kicked. How fast must he run in order to catch the ball before it hits the ground? (Sample Problems) Solution>>>
Solution to Problem No. 5 We have a projectile motion problem (as far as the football is concerned). Hence we have drawn a figure, chosen a CS, and write down the initial conditions (initial position & velocity) of the football (at t=0). x0 = 0; y0 = 0; v0x = v0 cos 37; v0y = v0 sin 37 The acceleration is: ax = 0; ay = - 32 ft/sec2. The general equations of motion for constant acceleration in 2-dimensions are: x(t) = (1/2) ax t2 + vox t + xo y(t) = (1/2) ay t2 + vox t + yo vx(t) = ax t + vox vy(t) = ay t + voy We insert the known values for acceleration & initial conditions and obtain the specific equations for the football: x(t) = (48)(4/5) t y(t) = - (1/2)(32) t2 + (48)(3/5) t vy(t) = - 32 t + (48)(3/5)
We can now answer any question regarding the motion of the football. In particular, we are interested in when it hits the ground (call this t). We have: y(t) = 0 = - 16 t2 + (48)(3/5) t t = 0, or t = 1.8 sec. Hence the ball will land at x(t) = x(1.8s) = (48)(4/5)(1.8) = 69 ft from the origin. We can now consider the 2nd player. His initial position (t=0) is 100 ft from the origin, and he must reach a point 69 ft from the origin in 1.8 sec if he is to catch the ball. Thus from the definition of average velocity, vave = (x2 - x1)/(t2 - t1) = (69 - 100)/(1.8) = - 17 ft/sec. The negative sign indicates that he must run toward the origin (negative x direction). (Sample Problems)
Prepared by: Amonoy, Ginger A. Bisnar, Diwata R. Gonzales, Charles Jourdan V. Lacayanga, Moises Jerome D. Leoncio, Krized Noviem M. Santos, Ruth Margarette L. BSCE 3-B