A particle is drop from a vertical height (h) and falls freely for a time (t) with the aid of a sketch explain how (h) varies with
(i) t
(ii) t*2
To explain how the vertical height (h) varies with time (t) and with time squared (t^2), let's consider the motion of a freely falling object under the influence of gravity.
(i) Variation of h with t:
When a particle is dropped from a vertical height and falls freely, its vertical position, or height (h), changes with time (t). In the absence of any other forces, an object falling freely near the surface of the Earth experiences a constant acceleration due to gravity, denoted as "g" (approximately 9.8 m/s^2).
Initially, when the particle is dropped, its height decreases, and as time elapses, the height decreases further. The relationship between height and time can be visualized using a sketch as follows:
^
h |
|
|
|______________________
/ ^
/ |
/ |
/ t
In the sketch, the vertical axis represents the height (h) and the horizontal axis represents time (t). As time increases, the height decreases linearly, forming a straight line with a negative slope. The steeper the slope, the faster the object is falling.
(ii) Variation of h with t^2:
The variation of height (h) with the square of time (t^2) can be understood by considering the formula for the displacement of a freely falling object:
h = (1/2) * g * t^2
In this equation, the height (h) is directly proportional to the square of the time (t^2), with a constant factor of (1/2) * g. As time increases, the height increases non-linearly with the square of time. The relationship between height and time squared can be visualized using a sketch as follows:
^
h |
|
|
|_________
/ ^
/ |
/ |
/ t^2
In this sketch, the vertical axis represents the height (h) and the horizontal axis represents the squared value of time (t^2). As time squared increases, the height increases exponentially, forming a curve that becomes steeper as time squared increases.
Remember that these explanations are based on the assumption of no other forces acting on the particle apart from gravity.
To explain how the vertical height (h) varies with time (t) and time squared (t^2), let's consider the factors affecting the vertical motion of a freely falling particle.
First, when a particle falls freely under the influence of gravity, its motion is governed by the laws of kinematics. In this case, we assume that there is no air resistance.
(i) Variation of height (h) with time (t):
When a particle is dropped from a certain height, its height decreases as time progresses. Initially, the particle has a high height, and as time increases, the particle moves closer to the ground.
The relationship between the height (h) and time (t) can be illustrated with a sketch of a simple graph where the vertical axis represents height (h) and the horizontal axis represents time (t).
At the starting point (t=0), the height (h) is maximum, and as time progresses, the height reduces continuously. The graph will be a decreasing curve, starting at some positive value on the vertical axis and decreasing as time goes forward. The curve will approach zero as time approaches infinity.
(ii) Variation of height (h) with time squared (t^2):
When we consider the relationship between height (h) and time squared (t^2), the graph will be different.
The relationship between height and time squared is quadratic. The graph will start at some positive value on the vertical axis, and initially, the rate at which the height decreases is high. As time squared (t^2) increases, the height reduces at a decreasing rate. The graph will form a concave downward curve. Eventually, at some point, the height will decrease to zero.
It's important to note that the rate at which the height decreases is greater for t^2 than for t because t^2 grows faster as time progresses.
In summary, the variation of height (h) with time (t) is a decreasing curve, while the variation of height (h) with time squared (t^2) is a concave downward curve.