Video Transcript
Fill in the blank. The force acting on a mass varying
according to the function 𝑚 of 𝑡 is equal to five plus two 𝑡 kilograms and moving
with a constant velocity of four meters per second is blank.
Now, remember that the force acting
on a body with varying mass is equal to the rate of change of momentum, so that’s
𝑚d𝑣 by d𝑡 plus 𝑣d𝑚 by d𝑡. Since the velocity is a scalar,
we’ll be using the scalar equation of Newton’s second law for variable mass instead
of the vector form. The question has given us a
function for the mass and a value for the velocity. We have that 𝑚 is equal to five
plus two 𝑡 kilograms and 𝑣 is equal to four meters per second. We need to differentiate both of
these with respect to 𝑡. Starting with 𝑚, when we
differentiate the constant five, we’ll get zero. And when we differentiate the two
𝑡, we’ll get two. Hence, d𝑚 by d𝑡 is equal to
two. Now, we have that 𝑣 is equal to
four, which is a constant. So, when we differentiate it, we’ll
get zero.
We’re now able to substitute these
values into our equation for 𝐹. We have that 𝐹 is equal to five
plus two 𝑡 multiplied by zero plus four multiplied by two. Since the first time is all
multiplied by zero, this will disappear. And so, we are left with 𝐹 is
equal to eight newtons. And here, we have reached our
solution, which is that a body with an initial mass of five kilograms which
increases at two kilograms per second which is moving with a constant velocity of
four meters per second must have a constant force acting on it of eight newtons. We can also note here that the
initial mass of the body does not affect the force required to maintain this
constant velocity. The only thing that matters is the
rate of change of the mass.
