DP IB Physics: SL

A. Space, time and motion

A.1 Kinematics

DP IB Physics: SL A. Space, time and motion A.1 Kinematics Linking questions:
a)	How does the motion of a mass in a gravitational field compare to the motion of a charged particle in an electric field?
b)	How are the equations for rotational motion related to those for linear motion?
c)	When can certain types of problems on projectile motion be solved by applying conservation of energy instead of kinematic equations?
d)	How effectively do the equations of motion model Newton’s laws of dynamics?
e)	How does a gravitational force allow for orbital motion?
f)	How does the motion of an object change within a gravitational field?
g)	How does graphical analysis allow for the determination of other physical quantities? (NOS)

a) How does the motion of a mass in a gravitational field compare to the motion of a charged particle in an electric field?
Solution:
A charged particle moving in an electric field and a mass moving in a gravitational field both involve forces acting on objects because of fields, but they also differ significantly.
When applied to mass-containing objects, gravitational force is always attractive but weaker than electric force.
Electric force, which is far more powerful than gravity, works on charged things and may be either attracting or repulsive.
⇒ The motion of a charged particle in an electric field:
In an electric field, a charged particle feels a force that propels it forward. The charge of the particle and the field’s direction determine the force’s direction.
If the particle is originally moving and the field is uniform, its trajectory will be a parabola; if it is initially stationary, it will travel along or against the field lines.
Figure 1 Charged particle in an electric field
⇒ The motion of a mass in a gravitational field:
A mass in a gravitational field will accelerate towards the field’s center as a result of gravity’s pull. Regardless of mass, this acceleration is the same for all objects in the same gravitational field.
An object in a gravitational field will be drawn in the direction of the field’s source, such as the Earth. how quickly the velocity of the item changes.
The acceleration is almost constant in a constant gravitational field (g = 9.8 m/s²).
An object’s mass (m) times its acceleration (a) equals the force (F) exerted on it: F = ma. This becomes F= mg in the context of gravity, where g is the acceleration brought on by gravity.
Figure 2 The motion of an object toward the Gravity
b) How are the equations for rotational motion related to those for linear motion?
Solution:
With the substitution of their rotating equivalents, angular displacement, angular velocity, and angular acceleration, for important variables like displacement, velocity, and acceleration, the equations for rotational motion are similar to those for linear motion.
For instance, you may construct a comparable rotational kinematic equation when there is constant angular acceleration if you know a linear kinematic equation for constant acceleration.
Rotational motion Vs Linear motion

Linear motion	Rotational Motion	Description
S		Displacement-angular displacement
[math]v = \frac{ds}{dt}[/math]	[math]\omega = \frac{d\theta}{dt}[/math]	Velocity-angular velocity
[math]a = \frac{dv}{dt}[/math]	[math]\alpha = \frac{d \theta}{dt}[/math]	Acceleration- Angular acceleration
[math]F = ma[/math]	[math]τ = Iα[/math]	Newton’s 2^nd law of motion
[math]p = mv[/math]	[math]L = Iω[/math]	Momentum-angular momentum
[math]K.E = \frac{1}{2} mv^2[/math]	[math]K.E = \frac{1}{2} Iω^2[/math]	Kinetic energy-rotational kinetic energy

Kinematic Equations:

Linear equation	Angular equation
[math]s = vt[/math]	[math]θ = ωt[/math]
[math]v = u + at[/math]
[math]S = ut + \frac{1}{2}at^2[/math]	[math]\theta = \omega_0 t + \frac{1}{2} \alpha t^2[/math]
[math]v^2 = u^2 + 2as[/math]	[math]\omega^2 = \omega_0^2 + 2\alpha\theta[/math]

c) When can certain types of problems on projectile motion be solved by applying conservation of energy instead of kinematic equations?
Solution:
When determining final velocities or heights, the conservation of energy can be used to solve some projectile motion problems rather than kinematic equations.
This is especially useful in situations where initial and final velocities or heights are known, or when you are only interested in those final values and the time of flight is not a significant factor.
Figure 3 Projectile motion
If the time of flight or the time it takes to reach a certain point is not a primary concern, or if you don’t have enough information about time, conservation of energy can be a useful tool.
Energy conservation concepts work effectively in situations when the projectile is changing from potential to kinetic energy, such when a ball is hurled upward and achieves its maximum height.
Kinematic equations are typically more appropriate when calculating the horizontal distance, a projectile travel or the time it takes for it to reach a specific height.
Kinematic equations could be more suitable if you need to examine the connection between acceleration and displacement or if the acceleration is not constant, which is not usual in ideal projectile motion.
When you are concerned in the interactions between energy types (kinetic and potential) rather than the precise time or acceleration changes involved, conservation of energy is essentially a strong tool for projectile motion issues.
d) How effectively do the equations of motion model Newton’s laws of dynamics?
Solution:
When motion is limited to a straight line and acceleration is constant, Newton’s laws of dynamics are accurately modelled by the equations of motion.
An object’s motion under these particular conditions may be succinctly predicted using these equations, which link displacement, beginning and ultimate velocities, acceleration, and time.
They are constrained, nevertheless, in circumstances where motion is not linear or acceleration fluctuates.
⇒ Equation of motion:
[math]s = vt[/math]

[math]v = u + at[/math]

[math]S = ut + \frac{1}{2}at^2[/math]

[math]v^2 = u^2 + 2as[/math]

[math]S = \frac{(u + v)}{2} t[/math]

[math]s = vt[/math]
[math]v = u + at[/math]
[math]S = ut + \frac{1}{2}at^2[/math]
[math]v^2 = u^2 + 2as[/math]
[math]S = \frac{(u + v)}{2} t[/math]

Newton’s Laws of dynamics:
⇒ First Law:
An object remains at rest or in uniform motion unless acted on by a net force.
First law: Inertia:
When acceleration will be zero then the equations reduce to
[math] v = u \\ S = ut[/math]
This represents motion with constant velocity,
⇒ Second Law:
The equation of motion assumes a constant acceleration
So
F = ma (force causes acceleration)
This only holds it the net force is constant
⇒ Third Law: (Action and reaction)
Every action has an equal and opposite reaction
Figure 4 Every action has an equal reaction
e) How does a gravitational force allow for orbital motion?
Solution:
By supplying the required centripetal force, gravitational force makes orbital motion possible. An orbiting item is constantly drawn towards the centre of the bigger body by this attraction, but its inertia keeps it going ahead, creating a curving path.
The orbit’s shape and speed are determined by the equilibrium between the pull of gravity and the object’s inertia.
⇒ Orbital motion:
The movement of an item around a fixed point, such as a satellite orbiting a planet or a planet orbiting a star, is known as orbital motion. This occurs when the gravitational attraction of the central body and the object’s forward speed are balanced.
A steady inward (centripetal) force, frequently gravity, prevents the item from travelling in a straight path and causes this motion.
Figure 5 Orbital motion
Gravitational force acts as a centripetal force
[math] F = \frac{GMm}{r^2}[/math]
Gravitational force and centripetal force:
[math]\frac{GMm}{r^2} = \frac{mv^2}{r}[/math]
The we get
[math]v = \sqrt{\frac{GM}{r}}[/math]
f) How does the motion of an object change within a gravitational field?
Solution:
An object’s motion inside a gravitational field is defined as a steady acceleration in the direction of the mass producing the field.
A falling object’s velocity increases steadily as a result of this acceleration, which is commonly represented by the symbol “g” (acceleration due to gravity).
[math]g = \frac{F}{m} = \frac{GM}{r^2}[/math]
Acceleration:
An object’s acceleration is the main result of gravity. This indicates that when the item approaches the gravitational field’s center, its velocity increases with time
[math]a = g = \frac{GM}{r^2}[/math]
Change in Velocity:
A falling object’s velocity rises linearly with time. In particular, for every second it descends on Earth, the velocity rises by around 9.8 meters per second.
Distance Travelled:
When subjected to gravity, an object’s distance travelled is not linear. It rises in direct proportion to the time squared.
Figure 6 Gravitational field
g) How does graphical analysis allow for the determination of other physical quantities? (NOS)
Using the link between the graph’s axes and the computed slope and area under the curve, graphic analysis enables the identification of other physical values.
For instance, the area beneath a velocity-time graph indicates displacement, while the slope of a position-time graph indicates velocity.
Understanding the relationships between various physical variables and how they alter over time is another benefit of this approach.
Position-Time Graphs:
An object’s average velocity during a certain time period is shown by the slope of a position-time graph. The velocity is constant if the graph is a straight line. The velocity changes when the line is bent, and the slope at any given position indicates the velocity at that moment.
Figure 7 Distance – time graph
Graphs of velocity and time:
An object’s acceleration is shown by the slope of a velocity-time graph. A velocity-time graph’s displacement, or change in position, is shown by the area under the curve.
Figure 8 Velocity-time graph
Acceleration-Time Graphs:
The change in velocity is represented by the area beneath an acceleration-time graph.
Figure 9 Acceleration – time graph