Question

The force needed to keep a car from skidding on a curve varies directly as the weight of the car and the square of the speed and inversely as the radius of the curve. Suppose a 3,960 lb. force is required to keep a 2,200 lb. car traveling at 30 mph from skidding on a curve of radius 500 ft. How much force is required to keep a 3,000 lb. car traveling at 45 mph from skidding on a curve of radius 400 ft.?

Answer 1

Force Required to Prevent Skidding

For the first car:

The force
`\( F_1 \)` needed to keep a car from skidding on a curve is given by the formula:

`$\[F_1 = k \cdot \left( \frac{\text{Weight} \cdot \text{Speed}^2}{\text{Radius}} \right)\]$`

where:

-
`\( k \)` is the proportionality constant

- Weight = 2,200 lb (for the first car)

- Speed = 30 mph (for the first car)

- Radius = 500 ft (for the curve of the first car)

-
`\( F_1 \)` = 3,960 lb (force required for the first car)

To find the proportionality constant
`\( k \)`:

`$\[k = \frac{F_1}{\left( \frac{\text{Weight} \cdot \text{Speed}^2}{\text{Radius}} \right)}\]$`

`$\[k = (3960)/(\left( (2200 \cdot 30^2)/(500) \right)) = \boxed{1.0}\]$`

For the second car:

Given:

- Weight = 3,000 lb

- Speed = 45 mph

- Radius = 400 ft

The force
`\( F_2 \)` for the second car is calculated as:

`$\[F_2 = k \cdot \left( \frac{\text{Weight} \cdot \text{Speed}^2}{\text{Radius}} \right)\]$`

`$\[F_2 = 1.0 \cdot \left( (3000 \cdot 45^2)/(400) \right) = \boxed{15187.5 \text{ lb}}\]$`

Hence, a force of 15,187.5 lb is required to keep a 3,000 lb car traveling at 45 mph from skidding on a curve of radius 400 ft.