Our goal in the Projectile Motion Lab will be to measure the range R and height

Question

Our goal in the Projectile Motion Lab will be to measure the range R and height h of an object in free fall projectile motion as a function of launch angle and launch velocity, to compare those values to theoretical values and to determine whether or not our assumption is valid that the x and y motions can be analyzed independently.

1. When the only force acting on an object is the gravitational force we say the object is in free fall. Any object in free fall experiences a vertical acceleration of ay = g = 9.8 m/s2 and a horizontal acceleration of ax = 0, since no force acts in the horizontal direction. (We ignore air resistance for the time being.) From our discussions and demonstrations in class we expect that the free fall motion of an object in the horizontal (x) direction and vertical (y) direction can be analyzed independently.

2. Review Chapter 4.3 of your text on "Projectile Motion".

3. Assuming that ax = 0 write out appropriate equations of motion for free fall in the horizontal (x) direction. (Use the following as necessary: g, vx0 as the x component of the initial velocity and t as the time.).

X=R=

4. Assuming that ay = g, write out appropriate equations of motion for free fall in the vertical (y) direction. (Use the following as necessary: g, vy0 as the y component of the initial velocity and t as the time.)

h=

5. The initial velocity of a ball launched into free fall is:

vi = 14.10 m/s and i = 31°.

(a) Calculate the x and y components of the initial velocity.

vx0

vy0

(b) Calculate the total time of flight ttotal of the ball.
ttotal =  s

(c) Calculate the maximum height hmax of the ball.
hmax =  m

(d) Calculate the horizontal distance R (range) of the ball.
R =  m

vx0

vy0

Explanation / Answer

following as necessary: g, vy0 as the y component of the initial velocity and t as the time.)

h = vyo * t – ½ * g * t^2

Use the following equation to determine the vertical and horizontal components of the initial velocity.


Vertical = v * sin , Horizontal = v * cos

As the object rise to its maximum height, its vertical velocity decreases from its initial amount to 0 m/s at the rate of 9.8 m/s. If the initial and final heights are the same, this happens in one half of the total time. During the total time, the horizontal velocity is constant. Let’s use the following equation to determine one half of the total time.

vf = vi – g * t, vf = 0, vi = v * sin
0 = v * sin – g * t
t = v * sin ÷ g

Total time = 2 * v * sin ÷ g
To determine the maximum horizontal distance, multiply the initial horizontal velocity by the total time.

Range = v * cos * 2 * v * sin ÷ g = v^2/g * 2 * cos * sin
2 * sin * cos = sin 2
Range = v^2/g * sin 2
This is how the range equation is derived.

Let’s use these equations in the following problem.

The initial velocity of a ball launched into free fall is:
vi = 14.10 m/s and i = 31°.

(a) Calculate the x and y components of the initial velocity.

vx0 = 14.1 * cos 31
This is approximately 12.1 m/s.

vy0 = 14.10 * sin 31
This is approximately 7.26 m/s.

(b) Calculate the total time of flight of the ball.


Total time = 2 * 14.10 * sin 31 ÷ 9.8 = This is approximately 1.48 seconds.

(c) Calculate the maximum height hmax of the ball.
The easiest way to determine the maximum height is to use the following equation.

vf^2 = vi^2 + 2 * a * d
vf is the ball’s vertical velocity at its maximum height. This is 0 m/s. a = -9.8m/s^2

0 = (14.10 * sin 31)^2 + 2 * -9.8 * d
d = 2.7 meters

(d) Calculate the horizontal distance R (range) of the ball.

Range = v^2/g * sin 2
Range = 14.1^2/9.8 * sin 62
This is approximately 17.91 meters.

Maximum height = ½ * (vi + vf) * t
vf = 0
t = v * sin ÷ g = 14.10 * sin 31 ÷ 9.8

This is approximately0.741 seconds
Maximum height = ½ * 14.10 * sin 31 * 14.10 * sin 31 ÷ 9.8
This is approximately 2.69 meters. Now you know two ways of determining the maximum height.

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