# Worksheet: Power as the Rate of Work

In this worksheet, we will practice defining the power of a force as the derivative of the work done by the force.

Q1:

A particle is moving under the action of the force . Its position vector at time is given by the relation . Find the rate of the work done by the force at .

Q2:

A particle of unit mass is moving under the action of a force . Its displacement, as a function of time, is given by . Find the value of when .

• A60 power units
• B228 power units
• C476 power units
• D124 power units

Q3:

A body of mass 3 kg moves under the action of a force N. At time seconds, the velocity of the body is given by . Find, in terms of , the power of the force, .

• A W
• B W
• C W
• D W
• E W

Q4:

A body of mass 17 kg moves under the action of a force . Its position vector at time is given by the relation . Given that is measured in newtons, in meters, and in seconds, write an expression for the power of force at time .

• A W
• B W
• C W
• D W

Q5:

A particle of mass 4 g is moving under the action of two forces and , where and . The position vector of the particle is given as a function of time by , where and are constants. Determine, in ergs per second, the power at which the resultant force acts on the particle 8 seconds after the start of motion.

Q6:

A constant force , measured in dynes, is acting on a body. The displacement of the body, after seconds, is given by , where and are two perpendicular unit vectors. Given that the force was working at a rate of 35 erg/s when , and at 43 erg/s when , determine .

• A dynes
• B dynes
• C dynes
• D dynes

Q7:

A body of mass 5 kg is moving under the action of the force measured in newtons. Its position vector after seconds is given by Find the work done by the force over the interval .

Q8:

A particle is moving under the action of the force . Its position vector at time is given by the relation . Find the rate of the work done by the force at .

Q9:

A particle of mass 2 g is moving under the action of two forces and , where and . The position vector of the particle is given as a function of time by , where and are constants. Determine, in ergs per second, the power at which the resultant force acts on the particle 6 seconds after the start of motion.

Q10:

A body of mass 2 kg moves under the action of a force N. At time seconds, the velocity of the body is given by . Find, in terms of , the power of the force, .

• A W
• B W
• C W
• D W
• E W

Q11:

A body of mass 4 kg moves under the action of a force . Its position vector at time is given by the relation . Given that is measured in newtons, in meters, and in seconds, write an expression for the power of force at time .

• A W
• B W
• C W
• D W

Q12:

A particle of unit mass is moving under the action of a force . Its displacement, as a function of time, is given by . Find the value of when .

• A38 power units
• B188 power units
• C373 power units
• D4 power units

Q13:

A constant force , measured in dynes, is acting on a body. The displacement of the body, after seconds, is given by , where and are two perpendicular unit vectors. Given that the force was working at a rate of 71 erg/s when , and at 31 erg/s when , determine .

• A dynes
• B dynes
• C dynes
• D dynes

Q14:

A constant force , measured in dynes, is acting on a body. The displacement of the body, after seconds, is given by , where and are two perpendicular unit vectors. Given that the force was working at a rate of 6 erg/s when , and at 14 erg/s when , determine .

• A dynes
• B dynes
• C dynes
• D dynes

Q15:

A body of mass 3 kg is moving under the action of the force measured in newtons. Its position vector after seconds is given by Find the work done by the force over the interval .

Q16:

A body of mass 8 kg is moving under the action of the force measured in newtons. Its position vector after seconds is given by Find the work done by the force over the interval .

Q17:

The power of a machine is given by the relation , where is the time elapsed in seconds. Find the work done by the machine in the first 8 seconds.

Q18:

The power of an engine is given by hp, where is the time in seconds. Find the power of the engine when , the work done over the time interval , and the maximum power of the engine.

• A , ,
• B , ,
• C , ,
• D , ,

Q19:

The power of an engine at time seconds is given by . Find the work done by the engine between and .

Q20:

A car has mass 1,066 kg. At time seconds, its engine works at a rate of . Given that at the car’s speed is 78 km/h , find its speed at . Give your answer to the nearest m/s.

Q21:

Find the time taken for a car of 1,236 kg to reach a speed of 126 km/h, given that the car started from rest and that the power of the engine is constant and equal to 103 metric horsepower. Use .