A vehicle has an engine that is connected directly to a gear box. In the highest gear, a 2 inch radius gear is turning a 14 inch diameter gear. This gear is connected to the drive shaft. The drive shaft attaches to a wheel and a tire that measures 2 and 1/3 feet in diameter. If the engine is pushing at 2500 rpm, how fast is the vehicle moving in miles per hour? Round your answer to the nearest tenth of a mile per hour.
(hint: watch your measurements
2500 * 2*2/14 rev/min * 7/3 * 12 * π in/rev * 60min/hr * 1mi/(12*5280)in = 59.5 mi/hr
To calculate the speed of the vehicle, we need to consider the gear ratio and the rotational speed of the engine.
First, let's calculate the gear ratio. The gear ratio is the ratio of input (engine) speed to output (drive shaft) speed. In this case, the engine is connected to a 2-inch radius gear, which is turning a 14-inch diameter gear.
The circumference of a circle is given by the formula: C = 2πr, where C is the circumference and r is the radius. Converting the gear diameters to radii, we have:
Input gear radius = 2 inches
Output gear radius = 14 inches / 2 = 7 inches
Now, we need to convert the radii to the same unit, preferably feet, since the tire diameter is given in feet.
Input gear radius = 2 inches / 12 inches/foot = 1/6 foot
Output gear radius = 7 inches / 12 inches/foot = 7/12 foot
Next, let's calculate the gear ratio. The gear ratio is the inverse of the ratio of the input to the output gear radii:
Gear ratio = Output gear radius / Input gear radius
Plugging in the values:
Gear ratio = (7/12 foot) / (1/6 foot) = (7/12) / (1/6) = 14/12 = 7/6
Now, let's calculate the speed of the drive shaft. Since both gears are connected directly, the speed of the drive shaft is the same as the speed of the output gear (14-inch diameter gear).
The rotational speed of the engine is given as 2500 rpm (revolutions per minute).
To convert the engine speed (rpm) to the drive shaft speed (rpm), we can use the gear ratio:
Drive shaft speed (rpm) = Engine speed (rpm) * Gear ratio
Plugging in the values:
Drive shaft speed (rpm) = 2500 rpm * (7/6) = 2916.67 rpm (rounded to the nearest hundredth)
Now, to calculate the speed of the vehicle in miles per hour, we need to consider the circumference of the wheel/tire.
The circumference of a circle is given by the formula: C = 2πr, where C is the circumference and r is the radius. The tire is given to have a diameter of 2 and 1/3 feet, so the radius would be:
Tire radius = (2 and 1/3) / 2 feet = (7/3) / 2 feet = 7/6 feet
The speed of the vehicle can be calculated using the formula:
Speed (miles per hour) = (Drive shaft speed * Tire circumference) / (12 inches/foot * 5280 feet/mile)
Tire circumference = 2πr = 2π(7/6) feet
Plugging in the values:
Speed (miles per hour) = (2916.67 rpm * 2π(7/6) feet) / (12 inches/foot * 5280 feet/mile)
≈ 69.4 mph (rounded to the nearest tenth)
Therefore, the vehicle is moving at approximately 69.4 miles per hour.