# An object has a mass of #6 kg#. The object's kinetic energy uniformly changes from #540 KJ# to # 32 KJ# over #t in [0, 4 s]#. What is the average speed of the object?

The average speed is

The kinetic energy is

Therefore,

and,

The equation of the line is

So,

So,

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To find the average speed of the object, we need to calculate the total distance traveled by the object during the given time interval and then divide it by the total time. We can use the formula for kinetic energy:

[ KE = \frac{1}{2} mv^2 ]

First, we need to find the initial and final velocities using the given kinetic energies and the mass of the object.

For the initial kinetic energy:

[ 540 \text{ kJ} = \frac{1}{2} \times 6 \text{ kg} \times v_0^2 ]

[ v_0^2 = \frac{2 \times 540 \text{ kJ}}{6 \text{ kg}} ]

[ v_0^2 = 180 \text{ kJ/kg} ]

[ v_0 = \sqrt{180} \text{ m/s} ]

For the final kinetic energy:

[ 32 \text{ kJ} = \frac{1}{2} \times 6 \text{ kg} \times v_f^2 ]

[ v_f^2 = \frac{2 \times 32 \text{ kJ}}{6 \text{ kg}} ]

[ v_f^2 = 10.67 \text{ kJ/kg} ]

[ v_f = \sqrt{10.67} \text{ m/s} ]

Now, we can calculate the average speed using the formula:

[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} ]

Since the object's acceleration is uniform, we can use the average velocity to find the average speed:

[ \text{Average velocity} = \frac{v_0 + v_f}{2} ]

[ \text{Average speed} = \frac{2 \times \text{Average velocity}}{\text{Total time}} ]

[ \text{Average speed} = \frac{2 \times (\frac{\sqrt{180} + \sqrt{10.67}}{2})}{4 \text{ s}} ]

[ \text{Average speed} = \frac{\sqrt{180} + \sqrt{10.67}}{4} \text{ m/s} ]

[ \text{Average speed} \approx 6.96 \text{ m/s} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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