A 500 kg go-cart is traveling at 15 m/s when the driver slams on the brakes. The cart comes to a stop in 7 m. If the same braking force is applied, how far will the cart travel after braking if it were traveling at 32 m/s?

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To answer this question, we need to use the laws of motion and the concept of kinetic energy.

First, let's find the initial kinetic energy when the go-cart is traveling at 15 m/s. The formula for kinetic energy is:

Kinetic Energy = (1/2) * mass * velocity^2

Given that the mass of the go-cart is 500 kg and the velocity is 15 m/s, we can calculate the initial kinetic energy.

Initial Kinetic Energy = (1/2) * 500 kg * (15 m/s)^2
Initial Kinetic Energy = 1/2 * 500 kg * 225 m^2/s^2
Initial Kinetic Energy = 56,250 kg * m^2/s^2

Now, let's calculate the final kinetic energy when the go-cart stops in 7 m. The final kinetic energy is zero because the go-cart comes to a stop. Therefore, the change in kinetic energy is:

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
0 = Final Kinetic Energy - 56,250 kg * m^2/s^2
Final Kinetic Energy = 56,250 kg * m^2/s^2

Now, let's find the final velocity when the go-cart is traveling at 32 m/s. We can use the formula for kinetic energy and solve for velocity:

Final Kinetic Energy = (1/2) * mass * velocity^2

Given that the mass of the go-cart is 500 kg, we can rearrange the formula to solve for velocity:

Final Kinetic Energy = (1/2) * 500 kg * (32 m/s)^2
56,250 kg * m^2/s^2 = (1/2) * 500 kg * 1024 m^2/s^2
56,250 kg * m^2/s^2 = 256,000 kg * m^2/s^2
Final Kinetic Energy = 256,000 kg * m^2/s^2

Now, let's find how far the go-cart will travel after braking when it is traveling at 32 m/s. We can use the formula for kinetic energy and solve for distance:

Final Kinetic Energy = (1/2) * mass * velocity^2

Given that the mass of the go-cart is 500 kg, we can rearrange the formula to solve for distance:

Final Kinetic Energy = (1/2) * 500 kg * velocity^2
256,000 kg * m^2/s^2 = (1/2) * 500 kg * velocity^2
256,000 kg * m^2/s^2 = 250 kg * velocity^2
velocity^2 = (256,000 kg * m^2/s^2) / (250 kg)
velocity^2 = 1024 m^2/s^2
velocity = sqrt(1024 m^2/s^2)
velocity = 32 m/s

Now, we know that the velocity after braking is also 32 m/s. However, the cart starts at a distance of 0 m after braking. Therefore, the distance the go-cart will travel after braking when it is traveling at 32 m/s is 0 m.

In conclusion, the cart will not travel any further after braking when it is traveling at 32 m/s since it will come to a stop in 0 m after applying the same braking force.