![]() d d d d Īs □ is known, we can express the rate of change ofĭ d d d □ □ = □ ( 1 2 □ + 9 ) = 1 2. The rate of change of displacement is equal to the instantaneous velocity, Time is not given by the question, but the displacement as a function of time is given. The function for the rate of change of the velocity of the body with respect to The function for the mass of the body with respect to time as follows: The rate of change of mass of the body can be found by differentiating So the force acting on the body is found using Answerīoth the velocity and the mass of the body vary with time, Write an expression for the force acting on the body at time □. Its mass varies with time such that □ = ( 8 □ + 9 ) k g. Its displacement from a fixed point is given by □ = 6 □ + 9 □ m. Where both the mass and the velocity of the body vary with time.Įxample 2: Finding the Force Acting on a Body with Variable Mass at Any TimeĪ body moves in a straight line. Let us look at an example of a force acting on a body, Its uniform motion, only the rate of change of its mass. The mass of the body does not affect the force required to maintain On by a force of 8 N to maintain that constant velocity. To increase in mass by 2 kilograms per second must be acted Therefore, a body that is moving at a constant velocity The rate of change of mass can be substituted into the formula as follows: The value of which does indeed increase with increasing □.ĭifferentiating □ ( □ ) with respect to time gives The mass inĪs a function of time, □ ( □ ), is given by The question expresses the mass of the body as a function of time. ![]() The body is stated to have a constant velocity, soįor a force to act on a body to move the body with constant velocity and hence with zeroĪcceleration, the mass of the body must change over the time that the force acts on it. AnswerĪ force acting on a body produces a change of momentum of the body: When two variables change with respect to a third variable, the product rule can be used to differentiate the product of the two variables.Įxample 1: Finding the Force Acting on a Body with Variable Mass Moving at a Constant Velocityįill in the blank: The force acting on a mass varying according to the function □ ( □ ) = ( 5 + 2 □ ) k g and moving with a constant velocity of 4 m/s is. Then the force acting on the body must account for both the increase in velocity and the increase in mass. If a body increases in both velocity and mass, Suppose that a body changes in both velocity and mass with respect to time. Equivalently, for a body that moves uniformly when a force acts on it, the rate of change of momentum of the body is given by Which is the constant mass form of Newton’s second law of motion. If the mass of an accelerating body is constant, the rate of change of the momentum of the body is given by So the rate of change of either term may need to be considered. The mass and velocity terms are both bracketed as either term can be time varying or constant, ![]() Where □ is the mass of the body and □ is the velocity of the body.Įxpressing Newton’s second law of motion in terms of rate of change of momentum gives ![]()
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