Instantaneous Velocity

Key Questions

• What represents instantaneous velocity on a graph?

  Provided that the graph is of distance as a function of time, the slope of the line tangent to the function at a given point represents the instantaneous velocity at that point.

  In order to get an idea of this slope, one must use limits. For an example, suppose one is given a distance function #x = f(t)#, and one wishes to find the instantaneous velocity, or rate of change of distance, at the point #p_0 = (t_0, f(t_0))#, it helps to first examine another nearby point, #p_1 = (t_0+a, f(t_0+a))#, where #a# is some arbitrarily small constant. The slope of the secant line passing through the graph at these points is:

  #[f(t_0+a)-f(t_0)]/a#

  As #p_1# approaches #p_0# (which will occur as our #a# decreases), our above #difference quotient# will approach a limit, here designated #L#, which is the slope of the tangent line at the given point. At that point, a point-slope equation using our above points can provide a more exact equation.

  If instead one is familiar with differentiation, and the function is both continuous and differentiable at the given value of #t#, then we can simply differentiate the function. Given that most distance functions are polynomial functions, of the form #x = f(t) = at^n + bt^(n-1) + ct^(n-2) + ... + yt + z,# these can be differentiated using the power rule which states that for a function #f(t) = at^n, (df)/dt# (or #f'(t)#) = #(n)at^(n-1)#.

  Thus for our general polynomial function above, #x' = f'(t) = (n)at^(n-1) + (n-1)bt^(n-2) + (n-2)ct^(n-3) + ... + y# (Note that since #t = t^1# (as any number raised to the first power equals itself), reducing the power by 1 leaves us with #t^0 = 1#, hence why the final term is simply #y#. Note also that our #z# term, being a constant, did not change with respect to #t# and thus was discarded in differentiation).

  This #f'(t)# is the derivative of the distance function with respect to time; thus, it measures the rate of change of distance with respect to time, which is simply the velocity.

  Psykolord1989 . · 1 · Aug 21 2014
• What is Instantaneous Velocity?

  Instantaneous velocity is the velocity at which an object is travelling at exactly the instant that is specified.

  If I travel north at exactly 10m/s for exactly ten seconds, then turn west and travel exactly 5m/s for another ten seconds exactly, my average velocity is roughly 5.59m/s in a (roughly) north-by-northwest direction. However, my instantaneous velocity is my velocity at any given point: at exactly five seconds into my trip, my instantaneous velocity is 10m/s north; at exactly fifteen seconds in, it's 5m/s west.

  CJ · · Sep 7 2014

• What is Instantaneous Velocity?
• How do you find the instantaneous velocity of a curve?
• How do you find the instantaneous velocity at #t=0# for the position function #s(t) = 6t^2 +8t#?
• How do you find the instantaneous velocity at #t=2# for the position function #s(t) = t^3 +8t^2-t#?
• How do you find the instantaneous velocity of the particle?
• How can instantaneous velocity be found from a displacement-time graph?
• How do you find instantaneous velocity in calculus?
• How does instantaneous velocity differ from average velocity?
• What is the difference between instantaneous velocity and speed?
• What represents instantaneous velocity on a graph?
• What is the derivative definition of instantaneous velocity?
• What is the instantaneous velocity of the ball at t=2 seconds if a ball is thrown in the air and its height from the ground in meters after t seconds is modeled by #h(t)=-5t2+20t+1#?
• What is the instantaneous velocity of an object moving in accordance to #f(t)=(12t,t^3)# at # t=1 #?
• What is the instantaneous velocity of an object moving in accordance to #f(t)=(12-t,t^2-t+7)# at # t=1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (2t-1,e^t-7)# at # t=1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(t^2),2t-e^t) # at # t=-1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(t^2),2t-te^t) # at # t=-1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(t-t^2),2t-te^t) # at # t=-1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(t-t^2),2e^t-t^2) # at # t=2 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(t-t^2),e^t/t^2) # at # t=2 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t^2,e^t-t^2) # at # t=3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (ln(t^2),3t-2) # at # t=2 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t/(t-5),3t-2) # at # t=-2 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t^3/(t-2),1/t) # at # t=1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t^3-t^2-3,e^(2t)) # at # t=6 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (tlnt,e^(2t)) # at # t=2 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t-e^t,e^(2t)) # at # t=2 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t-e^t,te^(2t)) # at # t=3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t/(e^t+2t),t) # at # t=4 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(sqrtt),1/t+2) # at # t=1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sqrt(t+2),t+4) # at # t=1 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sint,cost^3) # at # t=pi/2 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin^2t,cos3t) # at # t=pi/12 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin^2t,t-cos3t) # at # t=pi/4 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t^2sin(t-pi),tcost) # at # t=pi/3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (2sin(2t+pi),tcos2t) # at # t=pi/3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (2sin(2t+pi),t-sec(t/2)) # at # t=pi/3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (tsin(2t-pi/2),cost) # at # t=(5pi)/4 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(2t-pi/2),t) # at # t=(5pi)/4 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(2t-pi/2),cost/t ) # at # t=(3pi)/8 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin2tcost,tant-sect ) # at # t=(13pi)/12 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin2t-cos^2t,tantsect ) # at # t=(-pi)/12 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin2t-cos2t,sin(2t-pi/4)) # at # t=(-pi)/3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(t+pi),sin(2t-pi/4)) # at # t=(-pi)/3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(t+pi),cos(3t-pi/4)) # at # t=(pi)/3 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t^2,tcos(3t-pi/4)) # at # t=(pi)/8 #?
• What is the instantaneous velocity of an object moving in accordance to # f(t)= (t^2,tcos(t-(5pi)/4)) # at # t=(pi)/3 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= (tsqrt(t+2),t^2-2t) # at # t=1 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= ((t-2)^2,5t-3) # at # t=2 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= ((t-2)^3,sqrt(5t-3)) # at # t=2 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= (1/(t-4),sqrt(5t-3)) # at # t=2 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= (te^(t^3-t),t^2e^t) # at # t=1 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= (te^(t^2-3t),t^2-e^t) # at # t=3 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= (te^(-3t),t^2-te^t) # at # t=2 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= (t^2-8,tsqrt(t^2+3t-4)) # at # t=2 #?
• What is the instantaneous velocity of an object with position at time t equal to # f(t)= (sqrt(t^3-3t),tsqrt(t^2-2)) # at # t=3 #?
• The displacement (in meters) of a particle moving in a straight line is given by s = 2 t^3 where t is measured in seconds. How do you find the average velocity of the particle over the time interval [ 5 , 7]?
• An object moves according to the equation #y=1/(t+2),y>=0#, how do you find the velocity as a function of t?
• A particle moves according to the equation #y=t^4#, how do you find the velocity as a function of t?