More Motion along a Line
Just as you can plot a Position – Time Graph, you can also plot a Velocity – Time Graph. For the example of the Uniform Motion,
Time/t | Velocity/ms^{-1} |
1 | 2 |
2 | 2 |
3 | 2 |
4 | 2 |
5 | 2 |
(Velocity/ms^{-1} along the vertical axis, Time/s along the horizontal axis) Not a very exciting looking graph as the particle is moving at a constant velocity of 2m/s, however…
If you needed to know the change in position between two times, e.g. t = 2 and t = 4…
The change in position is equal to the velocity (2m/s) X the time period (t_{4}-t_{2}) = 4ms^{-1}s = 4m, i.e it’s the area under the graph! Often referred to as the signed area under the graph. Not very interesting? But what will happen when the velocity isn’t constant? What if it’s changing? 😉 Same applies, the change in position is the signed area under the graph. The Instantaneous velocity (at time t) is the gradient of the position-time graph at time t. It also follows that the Instantaneous speed at time t is v(t) = |v_{x}(t)|, i.e v of t is equal to the magnitude of v sub x (i.e. in the x direction) of t.
I always find it disturbing that in ‘real’ life how we can take uniform motion for granted and yet we are thrown around by non-uniform motion. Think of a train journey, along a straight track and at a steady speed, you could easily think you were stationary. Everyone has had that experience when the train pulls out of the station, is it us that’s moving or is it the train next door? Blank out the windows and would you know? Not until the velocity changes. The train accelerates by getting faster and you’re thrown into your seat, or slower and you’re thrown forward, or goes round a corner and you feel yourself being pushed sideways. I find that fascinating. Acceleration forces you to sit up and take notice of the real world outside of yourself. Why? Why not steady velocity? The everyday giving us clues about the nature of the Universe.
Acceleration is the rate of change of velocity and Instantaneous Acceleration is the rate of change of velocity with respect to time at time t. The units of acceleration must therefore be ms^{-1}s^{-1} = ms^{-2}
Time/t | Velocity/ms^{-1} |
1 | 1 |
2 | 4 |
3 | 9 |
4 | 16 |
5 | 25 |
The graph of an accelerating body (its velocity is not constant, it is changing – you’d be pushed back into your seat)
When you have two values (e.g. position and time, or velocity and time) that are related to each other, the relationship is called a function, e.g. if x = At + B, then x is a function of t, x(t).
If the value of a quantity f
is determined by the value of another quantity y, then f is a function of y, written f(y).
Examples of functions are:
Squares
f(y) = y^{2}
Cubes
f(y) = y^{3}
Polynomials such as;
Linear functions
f(y) Ay + B
Quadratic Functions
f(y) = Ay^{2} + By + C
Cubic Functions
f(y) = Ay^{3} + By2 + Cy + D
Given a function f(y) it is often possible to determine a related function of y, called the derived function, with the property that, at each value of y, the derived function is equal to the gradient of the graph of f against y. The derived function is referred to as the derivative of f with respect to y and may be represented by the symbol df/dy or df(y)/dy. When the function is a function of time, e.g. x(t), if you are like me (and lazy) then the derivative can be written as an x with a dot above it ẋ. (The dot
notation was created by Newton. The crazy dy/dt notation by Leibniz.)
Some simple Derivatives
Function f(y) | Description | Derivative df/dy | Example |
f(y) = A | A constant | df/dy = 0 | f(y) = 6
df/dy = 0 |
f(y) = y^{n} | A power of y | df/dy = ny^{n-1} | f(y) = y^{3}
df/dy = 3y^{2} |
f(y) = Ay^{n} | A constant times a power of y | df/dy = nAy^{n-1} | f(y) = 2y^{3}
df/dy = 6y^{2} |
f(y) = g(y) + h(y) | A sum of functions | df/dy = dg/dy + df/dy | f(y) = 2y^{3} + 4y
df/dy = 6y^{2} + 4 |
f(y) = Ag(y) | A constant times a function | df/dy = A(dg/dy) | f(y) = 0.5(2y^{3}+4y)
df/dy = 0.5(6y^{2} + 4) |
And so remembering that the instantaneous velocity of a particle at time t is given by the gradient of the position – time graph;
v_{x}(t) = dx(t)/dt
And similarly;
a_{x}(t) = dv_{x}(t)/dt = d^{2}x(t)/dt^{2} (the second derivative of x(t)) (or in my lazy way, acceleration a = ẍ 🙂 )
A special case of motion (just like the uniform motion) is the uniformly accelerated motion and as you might suspect the acceleration – time graph for this is just a straight line.
The equations for uniformly accelerated motion are:
s_{x} = u_{x}t + ½ a_{x}t^{2}
v_{x} = u_{x} + a_{x}t
v_{x}^{2} = u_{x}^{2} + 2a_{x}s_{x}
s_{x} = ½ (v_{x} + u_{x})t