OEF physical integral --- Introduction ---

This module actually contains 9 exercises on physical applications of definite integrals of one variable: force, energy, electronics, speed and acceleration, etc.

There are other modules of exercises on definite integrals: OEF definite integral on the theory and computation of definite integral, and OEF geometric integral for geometrical applications.

Constant acceleration

A car starts up with constant acceleration. At the end of seconds, the speed reaches kilometers per hour. What is during these seconds (in )?

Voltage on capacitor

A current milli-amperes enters into a capacitor of microfarads, where denotes the time in seconds. Compute the voltage on the capacitor (in volts, as a function of ), given that when , this voltage is zero.

Voltage on capacitor II

A current milli-amperes enters into a capacitor of microfarads, where denotes the time in seconds.

Given that the voltage on the capacitor is volts at the time , what is this voltage (in volts) when seconds?

Current into a solenoid

A voltage of volts is applied to a solenoid of milli-henris, where denotes the time in seconds. Compute the current entering into the solenoid (in amperes, as a function of ), given that when , this current is zero.

Current into a solenoid II

A voltage of volts is applied to a solenoid of milli-henris, where denotes the time in seconds.

Given that the current entering into the solenoid is amperes at the time , what is this current (in amperes) when seconds?

Rocket

A space rocket Orion X weighs tons at the starting point, in which tons are fuel. The engine of the rocket delivers a constant thrust of tons, and uses tons of fuel per minute. We want to know how high the rocket can go.

We suppose that the gravity force stays constant during the travel at 9.8 , and that the rocket goes always vertically.

Step 1. The combustion lasts seconds.

Spherical reservoir

A factory has a water reservoir under the form of a ball of meters of (internal) diameter. The usual water level is of meters above the bottom of the reservoir. One day, due to a system breakdown, this level has dropped to meters. How many cubic meters of water should be pumped into the reservoir, in order to get it back to the usual level?

Electrical resistance

A resistance of ohms is connected to an electric charge of volts, where t denotes the time in seconds.

Compute the average electric power dissipated on the resistance (in watts).

Spring

We suppose that the deformation of a spring is proportional to the force applied to it.

Suppose that a force of pulling at a spring lengthens it by . How much energy with respect to its rest position? (in kilogramme meters).

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