Question Statement

The tyre of a car is of radius 15 inches and it revolves 1000 times in a minute. What is the speed of the car in $\frac{km}{h}$ ?

Background and Explanation

This problem requires converting between imperial and metric units (inches to kilometers) and relating rotational motion to linear speed using the geometry of a circle. You will need to calculate the distance traveled in one revolution (circumference) and scale it up to find the hourly speed.

Solution

First, convert the radius from inches to meters. Since $1 inch = 2.54 cm$ :

$r = 15 inches = 15 \times 2.54 cm = 38.1 cm = 0.381 m$

Next, calculate the distance covered in one revolution. This equals the circumference of the tyre:

$Distance per revolution = 2 π r = 2 π (0.381) \approx 2.3939 m$

Now, calculate the total distance covered in 1000 revolutions (which occurs in one minute):

$Distance per minute = 1000 \times 2.3939 = 2393.9 m$

Thus, the speed of the car is $2393.9 m/minute$ .

Finally, convert this speed to $km/h$ . To do this, convert meters to kilometers (divide by 1000) and minutes to hours (multiply by 60, or equivalently divide by $\frac{1}{60}$ ):

$Speed = \frac{\frac{2393.9}{1000}}{\frac{1}{60}} km/h = \frac{2393.9 \times 60}{1000} km/h = \frac{143634}{1000} km/h = 143.63 km/h$

Key Formulas or Methods Used

Circumference of a circle: $C = 2 π r$
Unit conversion: $1 inch = 2.54 cm = 0.0254 m$
Speed conversion: $m/min \to km/h$ requires multiplying by $\frac{60}{1000} = 0.06$ (or dividing distance in meters by 1000 to get km, then dividing time in minutes by 60 to get hours)

Summary of Steps

Convert the radius from inches to meters: $r = 15 \times 0.0254 = 0.381 m$
Calculate the circumference (distance per revolution): $C = 2 π r \approx 2.3939 m$
Calculate distance traveled per minute: $1000 \times 2.3939 = 2393.9 m/min$
Convert to km/h: $\frac{2393.9 \times 60}{1000} = 143.63 km/h$

Solution

First, convert the radius from inches to meters. Since

1 inch = 2.54 cm

r = 15 inches = 15 \times 2.54 cm = 38.1 cm = 0.381 m

Next, calculate the distance covered in one revolution. This equals the circumference of the tyre:

Distance per revolution = 2 π r = 2 π (0.381) \approx 2.3939 m

Now, calculate the total distance covered in 1000 revolutions (which occurs in one minute):

Distance per minute = 1000 \times 2.3939 = 2393.9 m

Thus, the speed of the car is

2393.9 m/minute

Finally, convert this speed to

km/h

. To do this, convert meters to kilometers (divide by 1000) and minutes to hours (multiply by 60, or equivalently divide by

\frac{1}{60}

Speed = \frac{\frac{2393.9}{1000}}{\frac{1}{60}} km/h = \frac{2393.9 \times 60}{1000} km/h = \frac{143634}{1000} km/h = 143.63 km/h

7.10 Q-2

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

7.10 Q-2

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps