Two banked curves have the same radius. Curve A is banked at 11.0 °, and curve B is banked at an angle of 17.4 °. A car can travel around curve A without relying on friction at a speed of 15.3 m/s. At what speed can this car travel around curve B without relying on friction?
60 mph, i think. i first found the radius (since it's the same for both) then calculated the speed for curve b
Yeah i wonder if Den knows that explanation was NO help what so ever.
r of A = r of B
θ of A = 11°
θ of B = 17.4°
v of A = 15.3 m/s
v of B = ?
tan(θA) = (vA^2)/(rg)
derive to: r = (vA^2)/(tan(θA))(g)
r=(15.3^2)/(tan11)(9.8)
r=122.8866 m
tan(θB) = (vB^2)/(rg)
derive to: vB=sqrt[(r)(tanθB)(g)]
vB=sqrt[(122.8866)(tan17.4)(9.8)]
vB=19.4268 m/s
Answer: vB = 19.43 m/s
To find the speed at which the car can travel around curve B without relying on friction, we need to use the concept of centripetal force.
The centripetal force required to keep an object moving in a curved path is provided by the net force acting towards the center of the circle. On a banked curve, this net force is provided by the horizontal component of the normal force (N).
The formula for centripetal force (Fc) is:
Fc = (mv^2) / r
Where:
m is the mass of the object
v is the velocity of the object
r is the radius of the curve
To find the speed at which the car can travel around curve B, we need to determine the net force acting on the car in the horizontal direction on curve B.
The net force can be broken down into two components:
1. The force due to gravity (mg) acting vertically downwards.
2. The horizontal component of the normal force (N) acting towards the center of the curve.
Since curve B is banked at an angle, we can resolve the normal force into its vertical (Nv) and horizontal (Nh) components:
Nv = N * cosθ
Nh = N * sinθ
Where:
θ is the angle of banking for curve B (17.4°)
Since the car can travel around curve B without relying on friction, the net force in the horizontal direction is equal to the centripetal force:
Nh = Fc
Substituting the equation for centripetal force, we get:
N * sinθ = (mv^2) / r
Rearranging the equation to solve for v:
v^2 = (Nr * sinθ) / m
Taking the square root of both sides:
v = sqrt((Nr * sinθ) / m)
Now, let's substitute the values given in the question:
θ = 17.4°
v = 15.3 m/s (speed on curve A)
N = mg (since the car is not relying on friction, the normal force is equal to the weight of the car)
r (radius) is the same for both curves
Plugging in these values and solving for v:
v = sqrt((mg * r * sinθ) / m)
v = sqrt(g * r * sinθ)
Thus, the speed at which the car can travel around curve B without relying on friction is equal to the square root of (g multiplied by r multiplied by sine of 17.4°), where g is the acceleration due to gravity and r is the radius of the curve.
The value of g can be assumed as 9.8 m/s^2, and the radius of the curve is not given in the question. Plug in the values to calculate the speed at which the car can travel around curve B without relying on friction.