a car of height H could negotiate a bend on a banked road of certain degree and of radius r with a maximum speed. is it possible for a truck of height 3H to negotiate that same bend with same speed at that same angle.show proof to back the condition
To determine whether a truck of height 3H can negotiate the same bend on a banked road with the same speed as a car of height H, we need to consider the forces acting on the vehicles.
1. For the car:
Let's assume the maximum speed at which the car can negotiate the bend is v. At this speed, the car experiences two main forces:
- The gravitational force acting vertically down (mg), where m is the mass of the car and g is the acceleration due to gravity.
- The normal force (N) exerted by the banked road, which has both horizontal and vertical components.
The vertical component of the normal force (Nv) counters the gravitational force (mg) and can be calculated using trigonometry:
Nv = mg * cosθ,
where θ is the angle of the banked road.
The horizontal component of the normal force (Nh) provides the centripetal force required to keep the car moving in a circular path. It can be calculated as:
Nh = mg * sinθ.
The maximum speed v at which the car can negotiate the bend without skidding can be determined using the centripetal force equation:
Nh = (m * v^2) / r,
where r is the radius of the bend.
2. For the truck:
Now, let's consider the truck of height 3H. Since the truck is taller, it will experience different forces compared to the car.
The gravitational force acting vertically down (3mg) will be three times that of the car. However, the normal force and its components remain the same as in the case of the car.
Using the same equations for the truck, we can calculate the maximum speed (v') at which the truck can negotiate the bend without skidding. The equations will be:
Nv' = 3mg * cosθ,
Nh' = 3mg * sinθ,
Nh' = (3m * v'^2) / r.
To prove that the truck can also negotiate the bend at the same speed as the car, we need to compare the two maximum speeds (v and v').
Using the equations for Nh and Nh', we can equate them:
mg * sinθ = (3m * v'^2) / r.
Canceling the masses and rearranging the equation, we get:
v' = sqrt(v^2 / 3).
As you can see, the maximum speed for the truck (v') is equal to the maximum speed of the car (v) divided by the square root of 3. This means that for a truck of height 3H to negotiate the same bend on the banked road with the same speed as a car of height H, the maximum speed of the truck must be reduced by a factor of sqrt(3).
Therefore, it is possible for a truck of height 3H to negotiate the same bend with the same speed at the same angle, but the maximum speed of the truck needs to be reduced in comparison to the car.