Video Transcript
An engine of mass 80 metric tons
initially at rest on a horizontal track starts to move with constant
acceleration. When its velocity is 84 kilometers
per hour, its engine’s power is 2,520 kilowatts. Given that the total resistance to
the engine’s motion is one one fortieth of its weight, determine the magnitude of
its acceleration. Take 𝑔 equal to 9.8 meters per
second squared.
The question tells us that there is
constant acceleration. And we know there is constant
resistance as the weight stays the same. We will also assume a constant
force applied by the engine. We can, therefore, use two
equations. Firstly, power is equal to force
multiplied by velocity. Secondly, we’ll use the fact that
the sum of the net forces is equal to the mass multiplied by the acceleration. These are more commonly written as
𝑃 is equal to 𝐹 times 𝑉 and 𝐹 net is equal to 𝑚 times 𝑎.
We are given many of these values
in the question. However, we need to ensure that we
are in the standard units. For power, these are watts, for
forces, newtons, and for velocity, meters per second. The mass of the object, in this
case the engine, needs to be in kilograms and the acceleration in meters per second
squared. There are 1,000 kilograms in one
metric ton as 80 multiplied by 1,000 is 80,000, the mass of the engine is 80,000
kilograms. There are also 1,000 watts in a
kilowatt. This means that the power of the
engine when the velocity is 84 kilometers per hour is 2,520,000 watts.
There are 1,000 meters in a
kilometer and 3,600 seconds in an hour. This means that to convert from
kilometers per hour to meters per second, we multiply by 1,000 and then divide by
3,600. This is the same as dividing by
3.6. To calculate the velocity of the
engine in meters per second, we divide 84 by 3.6. This is equal to 70 over three or
seventy-thirds meters per second. We will now clear some space and
draw a diagram to model this situation.
The engine will have a downward
force equal to its weight. This will be equal to the mass of
80,000 kilograms multiplied by gravity 9.8 meters per second squared. This gives us a downward force
equal to 784,000 newtons. There will be a normal reaction
force acting vertically upwards, a resistance force acting against the direction of
motion, and a positive force driving the engine forward. We are told that the total
resistance is equal to one one hundred and fortieth of the engine’s weight. Multiplying one one hundred and
fortieth by 784,000 gives us 5,600. The resistance to the engine’s
motion is 5,600 newtons.
As power is equal to force
multiplied by velocity, we know that 2,520,000 is equal to the force multiplied by
seventy-thirds. Dividing both sides of this
equation by seventy-thirds gives us 𝐹 is equal to 108,000. The force driving the engine
forward is 108,000 newtons.
We can now calculate the sum of the
net forces. We have 108,000 in the positive
direction and 5,600 in the negative direction. The sum of these net forces will be
equal to the mass of 80,000 kilograms multiplied by the acceleration. The left-hand side of the equation
simplifies to 102,400. We can then divide both sides of
this equation by 80,000, giving us 𝑎 is equal to 1.28. The magnitude of acceleration of
the engine is 1.28 meters per second squared.
