# Finding max velocity for a kart on a circular, banked track

• PhysicsNoob2
In summary, the car moves around the circle at a speed of 9.25 m/s due to the centripetal force of static friction.
PhysicsNoob2
Homework Statement
You create a banked track of 12 degrees, now what's the maximum linear velocity?
And how would this change if a child was driving the kart instead?
Relevant Equations
F=mv^2/r
F=mg sin(theta)
This is a UK A-Level question that I'm really struggling with, and can't seem to find any resources online that explain it well.

I've been given the following details:
mass of gokart + driver = 520kg
Maximum frictional force between tyres and road on flat track F = 20% weight of kart+driver (104kg)
Bank of track = 12 degrees

And I've calculated the following in earlier questions:
Coefficient of friction = F/N = 0.0204
Angular velocity (Flat track) w = sqrt (F/mr) = 0.069 rad/s
Linear velocity (Flat track) v = sqrt (Fr/m) = 2.898 m/s

Here is my working out for the question I'm stuck on so far:
F (Force down the slope) = mg sin (theta)
F = 520 * 9.8 * sin(12)
F = 1059.51...

v = sqrt (Fr/m)
v = sqrt (1059.51... * 42 / 520)
v = 9.25 m/s (2dp)

I'm certain that I'm doing something wrong here, as this is using the force down the slope and not the centripetal force towards the centre of the track. But I've never seen anywhere that shows me how to calculate this? Or have I got this right and am worrying about nothing?

And when calculating a lighter kart (Taking in to account the child driving), I get the same maximum linear velocity. Is this correct? As intuition would suggest that the kart would move faster with a child in it than with an adult?

Last edited:
You need to draw a free body diagram (FBD) of the car as it goes around the circle. Note that the circle is horizontal which means that the net force on the car is horizontal. You are correct in that you should not use the downslope force as the centripetal force. Also, you completely ignored the force of static friction between the tires and the incline. It, also, has a horizontal component that contributes to the centripetal acceleration.

PhysicsNoob2 and Lnewqban
Yeah I've realised that I've made some big mistakes on the earlier bits (For example used mass rather than weight for the coefficient of friction calculations), so I'm going to go back and start again. I think I've got the idea now anyway, thanks for the pointers and I'll see how I get on! :)

## What factors determine the maximum velocity of a kart on a banked, circular track?

The maximum velocity of a kart on a banked, circular track is determined by several factors including the banking angle of the track, the radius of the circular path, the coefficient of friction between the kart's tires and the track surface, and the gravitational force acting on the kart. These factors collectively influence the centripetal force required to keep the kart on the track without slipping.

## How does the banking angle of the track affect the kart's maximum velocity?

The banking angle of the track plays a crucial role in determining the maximum velocity of the kart. A steeper banking angle allows for higher speeds because it helps to counteract the lateral forces acting on the kart, reducing the reliance on friction alone to keep the kart on the track. The optimal banking angle can be calculated using the formula: tan(θ) = v² / (r * g), where θ is the banking angle, v is the velocity, r is the radius, and g is the acceleration due to gravity.

## What is the role of friction in determining the maximum velocity on a banked track?

Friction between the kart's tires and the track surface provides the necessary grip to prevent the kart from slipping outwards due to centrifugal force. The coefficient of friction (μ) determines how much lateral force can be resisted before slipping occurs. The maximum velocity can be found by considering both the frictional force and the normal force components acting on the kart, using the equation: v_max = sqrt((r * g * (tan(θ) + μ)) / (1 - μ * tan(θ))).

## How can the radius of the circular track influence the maximum velocity of the kart?

The radius of the circular track directly affects the centripetal force required to keep the kart moving in a circular path. A larger radius requires less centripetal force for a given speed, allowing for higher maximum velocities. The relationship can be expressed as v_max = sqrt(r * g * tan(θ)), assuming an ideal scenario with no frictional losses. In practical scenarios, friction and other factors must also be considered.

## Can the kart's maximum velocity be increased by modifying the kart itself?

Yes, the kart's maximum velocity can be increased by making several modifications to the kart itself. Enhancements such as improving the tire quality to increase the coefficient of friction, lowering the kart's center of gravity to reduce the risk of tipping, and optimizing the aerodynamics to reduce drag can all contribute to higher maximum velocities. Additionally, ensuring the kart's weight distribution is balanced can improve handling and stability on the banked track.

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