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Find the time taken for a car of 1236 kilograms to reach a speed of 126 kilometres per hour, given that the car started from rest and that the power of the engine is constant and equal to 103 horsepowers.
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In this exercise, we’re told that the car’s mass is 1236 kilograms.
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We’ll call that value 𝑚.
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Its maximum speed achieved is 126 kilometres per hour.
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We’ll call that 𝑣 sub max.
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The power of the car’s engine is constant at 103 horsepowers.
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We’ll call that power capital 𝑃.
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We want to solve for the time it takes for the car to reach its maximum speed, given that it started from rest.
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We’ll call this time 𝑡.
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Let’s draw a sketch of this scenario to start.
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We have a car of mass 𝑚 which is initially at rest.
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But over some period of time, thanks to the power supplied by its engine, it achieves a speed of 𝑣 sub max.
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It’s that time 𝑡 that we want to solve for.
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And to start off solving for it, let’s recall the mathematical equation for power.
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Power 𝑃 is equal to the work done divided by the time it takes to do that work.
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And we can recall further that work through the work energy principle is equal to the change in kinetic energy that an object undergoes.
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Bringing these two relationships together, we can write that 𝑃 power is equal to the change in kinetic energy of our car over the time it takes for this change to occur or 𝑡 equals ΔKE over 𝑃.
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If we recall that an object’s kinetic energy is equal to one-half its mass times its speed squared, that means we can rewrite our equation for 𝑡 that it equals 𝑚, the car’s mass, times its maximum speed squared divided by two times 𝑃.
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This equation is true because the car’s initial kinetic energy since it was at rest is zero.
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Looking at our information given, we see that the mass of the car, its maximum speed, as well as the power of the engine are all supplied information.
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Before we plug in and solve for 𝑡, let’s convert the maximum speed, which is currently in units of kilometres per hour, to units of metres per second.
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And secondly, we’ll convert the power, which is in units of horsepower into units of watts.
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So starting with our maximum speed in units of kilometres per hour, we knew there is 1000 metres in each kilometer.
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So when we multiply by this fraction, we’re really multiplying by one.
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But it would change the unit since the units of kilometres cancel out.
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And we then do the same thing with hours and seconds.
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One hours equal to 3600 seconds.
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And our units of hours cancels out and we’re left with units of seconds for time.
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When we multiply these three fractions together, we find that our speed in units of metres per second is equal to 126 divided by 3.6.
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Next, we look at our power, which is currently in units of horsepower.
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There are approximately 745.7 watts in one horsepower.
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So when we multiply by this fraction, the units of horsepower cancel.
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And we’re left with units of watts for our engine power.
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Now that we’ve converted our speed and power into SI units, we’re ready to plug in and solve for 𝑡.
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When we enter these values on our calculator, we find that 𝑡 is approximately 10 seconds.
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That’s how long it would take this car to accelerate from rest to its given speed.