Question Video: Finding the Force Acting On a Body with Variable Mass Moving At a Constant Velocity Mathematics

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.