The essential part of the Hot Wheels speed booster is made of two fast spinning, rubberized disks of radius r=0.048 m each. The toy car entering the booster is in direct contact with both wheels, which send it away after brief encounter with it.
When the speed booster is set up horizontally at the edge of a table (h=0.92 m) it sends the car in the projectile motion so that it lands at horizontal distance of x=1.68 m away from the edge of the table.
Use this information to find the angular velocity of the spinning disks in the booster.
Give your answer in rpm -- revolutions per minute
To find the angular velocity of the spinning disks in the Hot Wheels speed booster, we can use the concept of projectile motion.
First, let's find the initial vertical velocity (v0y) of the toy car as it leaves the booster. The vertical distance the car travels is the height of the table, which is h = 0.92 m. We can use the equation for vertical motion:
s = ut + (1/2)gt²
where s is the vertical distance, u is the initial vertical velocity, g is the acceleration due to gravity, and t is the time of flight.
Since the car lands horizontally at a distance of x = 1.68 m, the time of flight can be calculated using the horizontal distance formula:
x = uxt
where ux is the initial horizontal velocity (which is constant throughout the projectile motion).
Since there is no horizontal acceleration, the initial horizontal velocity (uxt) remains constant throughout the motion. Therefore, we can rewrite the equation as:
t = x / ux
Now, let's find the initial vertical velocity (v0y):
0.92 = uyt - (1/2)gt²
Since the initial vertical velocity is zero (the car is in contact with the wheels), we can simplify the equation:
0.92 = -(1/2)gt²
Solving for t:
t² = (-2 * 0.92) / g
Now, let's calculate the value of g using the acceleration due to gravity (9.8 m/s²):
t² = (-2 * 0.92) / 9.8
Using a calculator:
t ≈ 0.429 seconds
Now, we can use the equation for horizontal velocity to find the initial horizontal velocity (ux):
x = uxt
1.68 = ux * 0.429
ux ≈ 3.92 m/s
The radius of each spinning disk is given as r = 0.048 m. The circumference of each disk is given by 2πr.
The distance traveled by the outer edge of each disk in one revolution (360 degrees) is 2πr.
The time taken for one revolution (T) can be calculated using the angular velocity:
T = 1 / (angular velocity)
To convert the angular velocity to revolutions per minute, we need to convert seconds to minutes and radians to revolutions. There are 2π radians in one revolution.
Now, let's calculate the angular velocity in radians per second:
ux = 2πr / T
T = 2πr / ux
T = 2π * 0.048 / 3.92
Now, let's convert the time taken for one revolution (T) to minutes:
T (in minutes) = T (in seconds) / 60
Finally, we can calculate the angular velocity in revolutions per minute (rpm):
angular velocity (rpm) = 1 / (T (in minutes) * 2π)
Substituting the values:
angular velocity (rpm) ≈ 1 / (T (in minutes) * 2π)