(a) At a certain instant, a particle-like object is acted on by a force F→=(4.0N)i^−(2.0N)j^+(9.0N)k^\vec{F} = (4.0\rm{N})\hat{i} - (2.0\rm{N})\hat{j} + (9.0\rm{N})\hat{k} while the object's velocity is v→=−(2.0m/s)i^+(4.0m/s)k^\vec{v} =- (2.0\rm{m/s})\hat{i} + (4.0\rm{m/s})\hat{k}. What is the instantaneous rate at which the force does work on the object? (b) At some other time, the velocity consists of only a y component. If the force is unchanged and the instantaneous power is -12 W, what is the velocity of the object?

(a) the instantaneous power is [Fill in the blank],
W

(b) the velocity is [Fill in the blank],
m/s

Question

(a) At a certain instant, a particle-like object is acted on by a force F→=(4.0N)i^−(2.0N)j^+(9.0N)k^\vec{F} = (4.0\rm{N})\hat{i} - (2.0\rm{N})\hat{j} +  (9.0\rm{N})\hat{k} while the object's velocity is  v→=−(2.0m/s)i^+(4.0m/s)k^\vec{v} =- (2.0\rm{m/s})\hat{i}  +  (4.0\rm{m/s})\hat{k}. What is the instantaneous rate at which the force does work on the object? (b) At some other time, the velocity consists of only a y component. If the force is unchanged and the instantaneous power is -12 W, what is the velocity of the object?

(a) the instantaneous power is [Fill in the blank],
 W

(b) the velocity is [Fill in the blank],
 m/s

BlackTom AI · Accepted Answer

We start by identifying that work rate (instantaneous power) is the dot product of force and velocity: P = F · v.

(a) Given F = (4.0, -2.0, 9.0) N and v = (-2.0, 0.0, 4.0) m/s, compute the dot product:
F · v = 4.0*(-2.0) + (-2.0)*0.0 + 9.0*4.0 = -8.0 + 0.0 + 36.0 = 28.0 W.
Thus, the instantaneous power (the rate at which the force does work on the object) is 28 W.

(b) At another moment, the velocity has only a y-component: v = (0, v_y, 0). The force remains F = (4.0, -2.0, 9.0) N. The instantaneous power is P = F · v = 4.0*0 + (-2.0)*v_y + 9.0*0 = -2.0 v_y.
They tell us P = -12 W. Setting -2.0 v_y = -12 gives v_y = 6.0 m/s. Since the power is negative, the velocity component along the force’s direction is such that it yields a negative projection; here that means the object is moving in the +y direction with a magnitude of 6 m/s. Therefore, the velocity is (0, 6, 0) m/s, i.e., 6 m/s in the +y direction.

类似问题

A car of mass 615.2 kg is driven along a straight flat road by an engine that provides a constant driving force. The car engine does 2.3 kJ of work while travelling at a constant speed 13.2 m/s over a time period of 3.3 seconds. What was the power developed by the car engine?

How many, if any, factors are necessary to determine power? 1.) Force exerted. 2.) Distance the force moves. 3.) Time required to do the work.

A child pulls on a wagon with a force of 10 N. If the wagon moves a total of 50 m in 5 s, what is the average power generated by the child, in watts?

A child pulls on a wagon with a force of 75 N. If the wagon moves a total of 42 m in 3 min, what is the average power generated by the child, in watts?

Work and power help us understand how force is applied across distance and how quickly that work is done over time (T or F)

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