You will need
- the ability to integrate
Let the body under the action of a certain force F has changed its velocity from V1 to V2 in a short period of time Δt. The acceleration of the body is equal to a=(V2-V1)/Δt, where a and V1 and V2 are vector quantities.
Substitute this expression into the formula for Newton's second law: F=ma=m(V2-V1)/Δt=(mV2-mV1)/Δt, not forgetting that the force F is also a vector quantity.
Write down the resulting formula in a slightly different form: FΔt=mΔV =Δp. Vector quantity FΔt, is the product of force at the time of its impact is called the impulse force and measured in Newtons multiplied by the second (N•s). And the product of mass and velocity p=mV – momentum of the body or body movement. This vector quantity measured in kilograms multiplied by meter per second (kg•m/s).
Thus Newton's second law can be formulated differently: the impulse of the force acting on a body is equal to change of momentum: FΔt=Δp.
If the impact force was very small, for example, during impact, the average force find: Fср=Δp/Δt=m(V2-V1)/Δt.Example: a Ball with a mass of 0.26 kg, flying at a speed of 10m/s. After hitting a player, the ball speed increased to 20 m/s Time of impact is 0.005 s. the Average force the hand of the player on the ball is equal in this case Fср=0,26•(20-10)/0.005=520Н.
If the force acting on a body is not constant, but changes with time according to the law F(t), by integrating the function F(t) at time t in the interval from 0 to T, find the change of momentum of the body: DP=F(t)dt.
And the formula Fср=dp/dt, determine the value of the average power.Example: a Force changing with time according to a linear law F=30t+2. Find the average impact force for 5s. First compute the momentum p=∫(30t+2)dt=15t2+2t , then the average force: Fср=(15t2+2t)/t=15t+2=15•5+2=77Н
Force is a vector quantity. If the result of the calculation value Fср turned out negative, it means that the force vector is directed in the direction opposite to the direction of the coordinate axis.
Don't forget when solving problems to translate everything used in formulas values in SI. Ie, mass in kilograms, speed in meters divided by the second, and force in Newtons.