The speed of a moving bullet can be determined by allowing the bullet to pass through
two rotating paper disks mounted a distance d apart on the same axle (Fig. below). From the angular
displacement of the two bullet holes in the disks and the rotational speed of the disks, we can
determine the speed v of the bullet. Find the bullet speed for the following data:
d = 80 cm, = 900 rev/min, and angular displacement = 31.0
φ=31⁰π/180⁰ = 0.54 rad.
n = 900 rev/min =900/60 =15 rev/s
t=φ/ω= φ/2πn=0.54/2π•15=0.006 s
v=d/t =0.8/0.006 = 133 m/s
Well, isn't this a bullet-ing situation! Let's calculate the bullet speed using some circus magic!
First, we need to convert the rotational speed from rev/min to radians/s. One revolution is equal to 2π radians, and a minute has 60 seconds. So, the rotational speed is (900 rev/min) × (2π radians/1 rev) × (1 min/60 s) = 30π radians/s.
Since the angular displacement is given in radians, we don't need to convert anything. It's already in the right units. So, the angular displacement is 31.0 radians.
Now, let's plug all these values into the formula:
v = d × ω
Where v is the speed of the bullet, d is the distance between the disks, and ω is the angular displacement divided by the rotational speed.
Plugging in the values, v = (80 cm) × (31.0 radians) / (30π radians/s).
Doing the math, we find that the bullet speed is approximately 26.55 cm/s.
So there you have it! The bullet is zooming along at approximately 26.55 centimeters per second. Stay clown-fident, my friend!
To find the speed of the bullet, we can use the following steps:
Step 1: Convert the rotational speed from revolutions per minute (rev/min) to radians per second (rad/s).
Given: ω = 900 rev/min
To convert rev/min to rad/s, we multiply by 2π/60:
ω = 900 rev/min * (2π rad/1 rev) * (1/60 min/s)
ω = 30π rad/s
Step 2: Convert the angular displacement from degrees to radians.
Given: θ = 31.0 degrees
To convert degrees to radians, we use the conversion factor π/180:
θ = 31.0 degrees * (π/180) rad/degree
θ = (31π/180) rad
Step 3: Calculate the linear speed of the bullet using the formula v = ω * d.
Given: d = 80 cm
To convert cm to m, we divide by 100:
d = 80 cm * (1 m/100 cm)
d = 0.8 m
Now, we can calculate the linear speed of the bullet:
v = ω * d
v = (30π rad/s) * (0.8 m)
v ≈ 75.398 m/s
Therefore, the speed of the bullet is approximately 75.398 m/s.
To find the speed of the bullet, we need to use the concept of angular velocity and the given data.
Angular velocity is defined as the change in angular displacement per unit time. It is represented by the Greek letter "omega" (ω).
First, we need to convert the rotational speed from revolutions per minute (rpm) to radians per second (rad/s). There are two important formulas to use for this conversion:
1 revolution = 2π radians
1 minute = 60 seconds
So, to convert from rpm to rad/s:
Angular velocity (ω) = (2π * rotational speed in rpm) / 60
Now, we have the angular displacement (θ) in radians and the angular velocity (ω) in rad/s. The relationship between angular displacement, angular velocity, and time is given by:
θ = ω * t
Here, θ is the angular displacement, ω is the angular velocity, and t is the time taken.
We need to rewrite this formula to solve for time:
t = θ / ω
Now, let's substitute the given values:
θ = 31.0 degrees = 31.0 * (π/180) radians (convert from degrees to radians)
ω = (900 rev/min) * (2π rad/1 rev) * (1 min/60 s) (convert from rpm to rad/s)
Plugging in the values, we can calculate:
t = (31.0 * (π/180)) / ((900 rev/min) * (2π rad/1 rev) * (1 min/60 s))
Now, we have the time taken (t) in seconds.
Next, we can use the formula for speed:
Speed (v) = Distance (d) / Time (t)
Substituting the given distance (d = 80 cm) and the calculated time (t), we can find the speed (v) of the bullet.
v = 80 cm / t
Calculate the final value to find the bullet speed.