6.1 Vector radius method of point motionM二、 Velocity17AArXr(t)Asshowninthethefigure,M→*displacement of the moving point M inr(t+ △t)Bthe time interval △t isMM' = △r = r(t+△t)-r(t)△rIt'scalledtheaverage1*velocity of the pointt△rdrThe velocity of the= lim * = lim1moving point at t instant:dtAt→0 △t△t-→>0That is, the velocity of a point is equal to the first derivative ofits vector diameter to time. The direction is along the tangentdirection of the trajectory
二、Velocity t r v = It’s called the average velocity of the point The velocity of the moving point at t instant: r dt dr t r v v t t = = = = → →0 0 lim lim That is, the velocity of a point is equal to the first derivative of its vector diameter to time. The direction is along the tangent direction of the trajectory. MM r r(t t) r(t) = = + − A O B M M r(t) r(t + t) r v v As shown in the figure, the displacement of the moving point M in the time interval t is 6.1 Vector radius method of point motion
6.1 Vector radius method of point motion三、 AccelerationAs shown in the figure, the change amount of the velocity vectorof the moving point M in the time interval △t is=MAvAiva*Average acceleration△tM'avaAcceleration of moving3point at t instant:Avdia = lim a* = lim=rdt△t->0△t-0 △tThat is, the acceleration of a point is equal to the firstderivative of its velocity to time and the second derivative of itsvector radius to time
三、Acceleration M M v v v v a a As shown in the figure, the change amount of the velocity vector of the moving point M in the time interval is t v v v = − t v a = Average acceleration Acceleration of moving point at t instant: v r dt dv t v a a t t = = = = = → →0 0 lim lim That is, the acceleration of a point is equal to the first derivative of its velocity to time and the second derivative of its vector radius to time. 6.1 Vector radius method of point motion
6.2 Cartesian coordinate method for themovement of points一、Equation of motionAs shown in the figure, establish a rectangularMcoordinate system onthereference body.Thenx=f(t) y=f(t) z= f(t)tK01This is the eguation of motion of a point in1yrectangular coordinates.XThe equation can be obtained by eliminating time t fromthe equation ofmotion:F(x,y,z)= 0It's called the trajectoryequation of the moving point
一、Equation of motion O x y z i j k r M x y z As shown in the figure, establish a rectangular coordinate system on the reference body. Then ( ) 1 x = f t ( ) 2 y = f t ( ) 3 z = f t This is the equation of motion of a point in rectangular coordinates. The equation can be obtained by eliminating time t from the equation of motion: F x y z ( , , ) 0 = It’s called the trajectory equation of the moving point. 6.2 Cartesian coordinate method for the movement of points
6.2 Cartesian coordinate method for themovement of points二、VelocityIt can be seen from the figure that theMvector radius of the moving point is:k= xi +yj+zk0idrdxdydzkvydtdtdtdtThen:.i+y.kVyi +y.dxdzxV.VxdtdtdtThat's the velocity of the point in terms of rectangular coordinates.That istheprojectionofthevelocityofa pointontherectangularcoordinateaxisisequal to the first derivative of the corresponding coordinate of a point withrespecttotime
二、Velocity O x y z i j k r M x y z It can be seen from the figure that the vector radius of the moving point is: r xi yj zk = + + k dt dz j dt dy i dt dx dt dr v = = + + v v i v j v k x y z = + + x dt dx vx = = y dt dy vy = = z dt dz vz = = That's the velocity of the point in terms of rectangular coordinates . That is, the projection of the velocity of a point on the rectangular coordinate axis is equal to the first derivative of the corresponding coordinate of a point with respect to time. Then: 6.2 Cartesian coordinate method for the movement of points
6.2 Cartesian coordinate method for themovement of pointsIf the projection of velocity is known, the magnitude ofvelocity isV= /x? + j? + z2xThe cosine of its direction is cos(v,i)= Vcos(,j) = v.Ncos(v,k)= =v
If the projection of velocity is known, the magnitude of velocity is 2 2 2 v = x + y + z The cosine of its direction is = = = v z v k v y v j v x v i cos( , ) cos( , ) cos( , ) 6.2 Cartesian coordinate method for the movement of points