In physics, the kinetic energy of a moving object is given by the formula K=1/2mv^2 where m is the mass of the object and v is the object's velocity. Suppose a rocket is increasing in velocity at 10m/sec^2, and is decreasing in mass at 15kg/sec because it is using up fuel. How fast is it's total kinetic energy changing when the velocity is 30m/sec and the mass is 1000kg? Round to the nearest tenth.

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Mathematics

2 September, 18:06

In physics, the kinetic energy of a moving object is given by the formula K=1/2mv^2 where m is the mass of the object and v is the object's velocity. Suppose a rocket is increasing in velocity at 10m/sec^2, and is decreasing in mass at 15kg/sec because it is using up fuel. How fast is it's total kinetic energy changing when the velocity is 30m/sec and the mass is 1000kg? Round to the nearest tenth.

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Slim · Answer 1

You need to take the derivative of both sides of the kinetic energy equation and then plug in the necessary values. The change in kinetic energy is dK/dt. If you take the derivative of the other side, you have to use product rule (since both m and v are variables). This gives dK/dt = (1/2) (dm/dt) (v^2) + (2) (1/2) mv (dv/dt)

= (1/2) (dm/dt) (v^2) + mv (dv/dt), where dm/dt is the rate of change of mass, and dv/dt is the rate of change of the velocity at the given time.

If you plug in for the given values (dm/dt = - 15, dv/dt = 10, m = 1000, v = 30), then you get dK/dt = (1/2) (-15) (30^2) + (1000) (30) (10) = - 6750 + 300,000 = 293,250 J/s. Notice that the answer is in units of Joules per second, beause you took the first derivative of kinetic energy with respect to time.