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