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Let's assume we have an object moving in the x-direction alone, calling its position x(t), where t is time.; and imagine it has constant acceleration a. Acceleration is the second derivative of time - in terms of calculus,
d2x/dt2 = a. (1)
So we can get the graph of the position by integrating (1). Doing this once gives us the velocity:
dx/dt = at + C
where, if the initial velocity (the velocity at time t = 0) is given by u and we substitute this in, we get
u = a*0 + C so that C = u, giving
dx/dt = at + u (2)
Now to get the solution for the position of the object against time, we integrate (2):
x(t) = (1/2)*at2 + ut + D
And, if the position of the object at time t = 0 is just called x(0), we get
x(0) = (1/2)*a*02 + u*0 + D so that D = x(0), finally giving
x(t) = (1/2)*at2 +ut + x(0) (3)
(3) you might recognise as the general equation of motion of an object with constant linear acceleration. Note that it is a quadratic function, and so is generally shaped like a parabola (See 'parabola', under wiki; [http://en.wikipedia.org/wiki/Parabola] especially the top-most line drawing (although imagine it upside down as the track of a ball thrown upwards, under the constant acceleration of gravity) and also the picture of the bouncing tennis ball at the bottom, each period in the air is an example of a parabolic track.)
Final point - note that the parabola (in projectile motion cases) -always- has a curve leading up from the ground and then back down. Sounds obvious, i know; but recall that objects moving under gravity are measured as having a -negative- acceleration, hence the first term of the quadratic is negative,the parabola will be upturned compared to a graph like x = t2. Hope this helps ;)
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How Can you tell from a velocity-time graph that the velocity is constant?
velocity is nothing but speed of a body in the given direction. suppose if body is moving with constant velocity then VT graph will be parallel to the X -axis, if not then the VT graph is not parallel to the X-axis it means then object is moving with different velocity or it has its dierection or both velocity and aswell as direction.
A graph showing zero slope for the position vs time always indicates that the object is not moving?
If the speed/time graph slops negatively, that's an indication that the speed is decreasing, i.e. the object is slowing down. The negative slop is also called negative acceleration, since acceleration is the rate of change of velocity.
What is the definition of velocity time graph?
A velocity-time graph shows how an object's velocity changes over time. The slope of the graph represents the object's acceleration, and the area under the curve represents the total displacement of the object. It is a useful tool for understanding an object's motion.
Which two graphs represents the motion of an object on which the net force is zero?
A horizontal line on a position-time graph or a stationary line on a velocity-time graph represents the motion of an object with zero net force. These graphs indicate constant velocity motion, where the object is moving at a consistent speed in a straight line.
What information can be learned from a position-time graph and from a speed-time graph?
A position time graph can show you velocity. As time changes, so does position, and the velocity of the object can be determined. For a speed time graph, you can derive acceleration. As time changes, so does velocity, and the acceleration of the object can be determined.If you are plotting velocity (speed) versus time, the slope is the acceleration.
