The purpose of this lab is to study the properties of projectile motion. From the motion of a steel ball projected horizontally, the initial velocity of the ball can be determined from the measured range. For a given initial velocity, the projectile range will be measured for various initial angles, and also calculated by applying the theory for motion with constant acceleration. For further background information, refer to the sections in your textbook on projectile motion and motion with constant acceleration. THEORY For a given initial velocity, v0 , and initial position, s0 ,the position of a particle, s, as a function of time, undergoing constant acceleration, a is given by sr = sr 0 + vr 0 t + 12 ar t 2( 1 ) This is a vector equation and can be broken up into its x, y, and z components. Since the motion is in a plane, we need only look at the x and y components. If we neglect air resistance, the acceleration in the y direction is -g, due to gravity. The acceleration in the x direction is zero. Hence, the vector equation (1) becomes two scalar equations: If we eliminate t in Eqs.(5) we get y as a function of x. gx2 and solving for vo we get

x = x0 + v 0x t(2) y=y+v t-1gt2

0 0y In terms of the angle θ, and the initial speed vo, the initial velocity components are v0x=v0cosθand v0y=v0sinθ A.For the case in which θ = 0° (initial velocity is horizontal), Eqs. (2) and (3) become x=vtand y=h-1gt2 0

2(3)

(4) 2(5)

y = h − 2v2(6) o

When the object hits the floor the x and y positions are x = R and y = 0. Hence, Eq. (6) becomes 0 = h − gR 2

1

Projectile Motion

2v2 o

❏ Spring gun set-up ❏ Gun mounted on frame w/ protractor ❏ Hook collar w/ pointer ❏ 2 Bench clamps ❏ 2 Aluminum bar supports w/ wing nut ❏ Short rod ❏ Spring gun ball ❏ Bar level ❏ Two-meterstick ❏ Plumb-bob ❏ Masking tape ❏ Paper

v 0 = Rgfor the caseθ = 0° 2h

(7) B.Now consider the case in which θ ≠ 0° (initial velocity is not horizontal). If we solve Eq. (2) for t and substitute the result into Equation 3, (using xo = 0 and yo = h) we get y=h+v0yx- g x2 We can use Eqs. (4) to rewrite this in terms of quantities you will measure: g

v0x2v 2 0x(8)

y = h + xtan θ - x2 2v02cos2θ (9)

Again, the range R is the value of x when y = 0; substituting this gives h + R tan θ - R 2g= 0 2v02cos2θ (10) This is a quadratic equation in R; solving for R we get the range equation, R = v02cosθ sin θ ±sin 2θ + 2hg

g

v 02

APPARATUS

EXPERIMENT

The experiment consists of measuring the range R of a small ball fired from a spring gun at various angles from the vertical. The apparatus (see Fig. 1) allows both the angle of projection and the initial velocity to be varied. (Note: Do not load or fire the spring gun until its use has been described and demonstrated by your instructor) The angle of projection is measured with a protractor, and the initial velocity can be varied by adjusting the tension on the spring with the adjusting knob on the back of the gun. After the instructor has demonstrated the use of the gun, load it and fire a few practice shots. Adjust the spring tension so that the range, when fired horizontally, is between one and two meters. Practice firing the gun the same way each time to minimize the variation in initial velocity. 2Projectile Motion

(11)

Two-meterstick Adjust knob

θ

x =0 y =0

Lock knob

yo = h

Recording Paper

x = R y =0

Figure 1: The spring gun setup.

PROCEDURE

1.Level the gun using the level provided. Tighten the knurled brass screw. Now adjust the pointer so that it indicates 0°. Leave the gun in this position and be sure all screws and clamps are tight. 2.Use a plumb bob to locate the point on the floor directly under the point where the ball leaves the gun. Mark this point on a piece of paper or tape fastened to the floor. Measure the height h of the ball from the floor. (Should you measure to the center of the ball or the bottom of it?) Note that this height changes as the angle is varied. 3....