to study the types of motion and equation of motion graphically in 500 words
Answers
While studying motion of bodies we have to keep two things in mind. We have to study various characteristics associated with moving bodies while we are to explore and study in detail, about the cause of producing motion. Kinematics is the branch of physics which deals only with the description of motion of bodies. Mainly there are two type of graphs are belonging to the kinematics. One is position-time graph and the other is velocity-time graph.
These graphs will have validity only if motion under study is along a straight line. Then, displacement, velocity and acceleration vectors are collinear and can be treated as algebraic quantities. Let x-axis be the path of motion. Then, x-coordinate represents the magnitude of position vector.
Position - Time GraphIf we plot time t along the x-axis and the corresponding position (say x) from the origin O on the y-axis, we get a graph which is called the position-time graph. This graph is very convenient to analyse different aspects of motion of a particle. Let us consider the following cases.
(i) In this case, position (x) remains constant but time changes. This indicates that the particle is stationary in the given reference frame. Hence, the straight line nature of position-time graph parallel to the time axis represents the state of rest. Note that its slope (tan θ) is zero.
(ii) When the x-t graph is a straight line inclined at some angle (θ≠) with the time axis, the particle traverses equal displacement Δx in equal intervals of time Δt. The motion of the particle is said to be uniform rectilinear motion. The slope of the line measured by Δx/Δt = tθ represents the uniform velocity of the particle.
(iii) When the x-t graph is a curve, motion is not uniform. It either speeds up or slows down depending upon whether the slope (tan θ successively increases or decreases with time. As shown in the figure the motion speeds up from t = 0 to t=t1 (since the slope tan θ increases). From t=t1 to t=t2, AB represents a straight line indicating uniform motion. From t=t2 to t=t3, the motion slows down and for t>t3 the particle remains at rest in the reference frame.
The adjacent figure shows the displacement-time graph of a particle moving on the x-axis. Choose the correct option given below.
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The Velocity - Time GraphThe velocity-time graph gives three types of information.
(i) The instantaneous velocity.
(ii) The slope of the tangent to the curve at any point gives instantaneous acceleration.
a = dv/dt = tan θ
(iii) The area under the curve gives total displacement of the particle.
Now, let us consider the uniform acceleration. The velocity-time graph will be a straight line.
The acceleration of the object is the slope of the line CD.
a = tan θ = BC/BD = (v-u)/t
v = u +at …... (1)
The total displacement of the object is area OABCD
s = Area OABCD = Area of rectangle OABD + Area of triangle BCD
s = ut + (1/2)at2 …... (2)
Again, s = Area OABCD
= 1/2(AC + OD) x OA = 1/2(v + u)xt
The acceleration-time graph:- Acceleration time curves give information about the variation of acceleration with time. Area under the acceleration time curve gives the change in velocity of the particle in the given time interval. Refer this simulation that shows a car traveling along a graph:-
The purpose is to show that, the car can traverse the graph, then it's first derivative is a continuous function.
DID YOU KNOWThe displacement can have positive, negative or zero value. The displacement is never greater than the actual distance travelled. A change in either speed or direction of motion results in a change in velocity. It is not possible for a particle to possess zero speed with a non-zero velocity. The velocity-time graph for a uniformly accelerated body is a straight line.