1. Motion in a straight line

• Felix Baumgartner broke the records for the highest altitude reached by a parachute jump.
• The highest manned balloon flight, and the greatest free fall velocity in October 2012.
• At a height of over 39 km, he leaped and attained a speed of [math]377 m.s^{-1}[/math] ([math]1357 km.h^{-1}[/math]).
• Careful planning and design are necessary for such a challenge to be successfully completed.
• Felix relied on physicists’ calculations using the equations of motion to predict his time of fall and to determine when it was safe to open his pressurized space suit.
• Felix also needed a balloon inflated to precisely the right pressure at sea level so that it could expand as the atmospheric pressure decreased at high altitude.
• Distance and Displacement
  • Distance: The total length of the path an object follows.
    Unit: meter
    Formula: Distance = speed * time
    Quantity: scalar
  • Displacement: The distance an object travels from its initial to its final position
    Unit: meter
    Formula:  [math] \vec{s} = \vec{s_f} \, \text{(final point)} – \vec{s_i} \, \text{(initial point)} [/math]
    Quantity: Vector
• Speed and velocity:
  • Speed: Speed is a scalar quantity that refers to how fast an object is moving.
    It is defined as the distance traveled per unit time (e.g., meters per second, miles per hour).
    Speed is a rate of motion, and it has no direction.
  • Velocity: Velocity is a vector quantity that refers to an object’s speed in a specific direction.
    It is defined as the displacement (change in position) per unit time.
    Velocity has both magnitude (amount of movement) and direction.
  • Acceleration: Acceleration is a vector quantity that measures the rate of change of velocity.
    It is defined as the change in velocity (Δv) per unit time (t).
    Acceleration is a change in the speed or direction of an object.
  • Types of acceleration:
    1. Uniform acceleration: When the acceleration is constant.
    2. Non-uniform acceleration: When the acceleration changes over time.
    3. Instantaneous acceleration: The acceleration at a specific instant.

⇒ Displacement-time graphs

• The terms distance and displacement are used interchangeably when the motion is in a straight line.

  
  Figure 1 Displacement-time graph for a walker

• Similarly, if the motion is in a straight line, then speed and velocity are also used interchangeably.
• In a displacement-time graph, time is taken along horizontal axis while vertical axis shows the distance covered by the object.
• • A walker’s displacement–time (s–t) graph is displayed in Figure 1.
• She advances to the right for the first three seconds, which is selected to be positive.
• After that, she stops moving for two seconds, maintaining her displacement.
• Finally, she moves back to the left for five seconds, ending up where she began.
• • The graph’s gradient may be used to determine the walker’s speed.
• Over the region AB:

[math] \text{Velocity} = \frac{\text{Change in displacement}}{\text{time taken}} \\
v = \frac{\text{final displacement} – \text{initial displacement}}{\text{time taken}} \\
v = \frac{(10 – 0) \, \text{m}}{3 \, \text{s}} \\
v = \frac{10 \, \text{m}}{3 \, \text{s}} \\
v = 3.3 \, \text{m/s} [/math]

• Over the region BC:
  No displacement cover in that region. So,
  v = 0
• Over the region CD:
• [math] v = \frac{\text{final displacement} – \text{initial displacement}}{\text{time taken}} \\
  v = \frac{(0 – 10)}{5} \\
  v = -2 \, \text{m/s} [/math]
• Because the gradient is negative over the region CD the velocity is negative.

  
  Figure 2 A displacement–time graph for a vehicle that is accelerating.

• There are three constant velocities shown in Figure 1. This indicates that the graphs are “straight line” since the graphs’ gradients are constant.
• A displacement–time graph of an accelerating vehicle is shown in Figure 2. The gradient rises with increasing velocity.
• We measure the gradient at point P by drawing a tangent to the curve there in order to get the velocity. We quantify a tiny displacement shift, Δ [math] \vec{s}  [/math] , that happened during a little time span, Δt.

⇒ Velocity–time graphs

• In a Velocity-time graph, time is taken along x-axis and velocity is taken along y-axis.
• Independent values along x-axis while dependent values along y-axis.
  • An airplane’s velocity–time graph during takeoff is seen in Figure 3.
• Part AB of the graph illustrates how the plane accelerates initially at a constant pace.

  
  Figure 3 airplane’s velocity–time graph during takeoff

• After then, the plane accelerates more slowly until it reaches point D, where its velocity is constant.
• The graph illustrates that the acceleration is 4 m/s across portion AB.
• The graph’s gradient is as follows: (80ms^-1)/(20s).
• However, the gradient is altering throughout the region BC, thus we now use the following formula:
• 
• Where Δ [math] \vec{v} [/math] means a small change in velocity, and Δt means a small interval of time. At point P:
• 
• Figure 4 shows how we can use a velocity–time graph to calculate the distance travelled by a motorbike.
• In this graph, the motorbike travels at a constant velocity over the period AB, before decelerating over the period BC. While the velocity is constant, the distance travelled is represented by area A1, which is 320m.
• While the bike decelerates, we could use the formula
• 
• So

  
  Figure 4 How to use a velocity–time graph to calculate the distance travelled by a motorbike.
  

• displacement= average velocity*time taken
• But the average velocity is 10ms−1, which is the average of [math]0ms^{-1} [/math] and [math]20ms^{-1} [/math].
• So, the distance can also be calculated using the area:
• 
• Figure 4 also shows that when the gradient of a velocity–time graph is zero, then the velocity is constant.
• When the gradient is negative, the bike is decelerating.
• The area under a velocity–time graph is the distance travelled.
• The gradient of a velocity–time graph is the acceleration.

2. Equations for uniform acceleration

• There are three basic equations of motion for bodies moving with uniform acceleration.
• These equations relate initial velocity, final velocity, acceleration, time and distance covered by a moving body.
• To simplify the derivation of these equations, we assume that the motion is along a straight line.
• Hence, we consider only the magnitude of displacements, velocities, and acceleration.
• •  Consider a body moving with initial velocity [math]\vec{v_i}[/math] in a straight line with uniform acceleration [math]\vec{a}[/math].
• Its velocity becomes [math]\vec{v}[/math] after time t.
• The motion of body is described by speed-time graph as shown in figure 5 by line AB.
• The slope of line AB is acceleration [math]\vec{a}[/math].
• The total distance covered by the body is shown by the shaded area under the line AB.
• Equations of motion can be obtained easily from this graph.

• First equation of motion
• Speed-time graph for the motion of a body is shown in figure 5. Slope of line AB gives the acceleration a of a body.

  
  Figure 5 Speed-time graph. Area under the graph
  gives the distance covered by the body.

• 
• Second equation of motion
• In speed-time graph shown in figure 5, the total distance S travelled by the body is equal to the total area OABD under the graph. That is
• Total distance(S)=Area of (Rectangle OACD = triangle ACB)     (2)
• 
• Third equation of motion
• The whole area OABD under the speed-time graph in figure 5 gives the total distance S traversed by the body.
• [math] \text{Total area } OABD = S = \frac{(OA + BD)}{2} * OD [/math]
• [math]2S = (OA + BD) * OD [/math]
• [math] \text{Multiply by } \frac{BC}{OD} \text{ on both sides:} [/math]
• [math]2S * \frac{BC}{OD} = (OA + BD)* OD * \frac{BC}{OD} [/math]
• Then
• [math] 2S * \frac{BC}{OD} = (OA + BD) * BC \qquad (4) [/math]
• We know: [math] \frac{BC}{OD} = \vec{a}, \, OA = \vec{v_i}, \, BD = \vec{v_f}, \, BC = BD – CD, \, \text{and } CD = OA = \vec{v_i} [/math]
• So, equation (4) becomes:
• [math]2S * \frac{BC}{OD} = (OA + BD) * (BD – CD) \\
  2\vec{a} S = (\vec{v_i} + \vec{v_f}) * (\vec{v_f} – \vec{v_i}) [/math]
• [math]2\vec{a} S = \vec{v_f}^2 – \vec{v_i}^2 \qquad (5)  [/math]

⇒ Acceleration due to gravity, g

• One of the first scientists to study the acceleration of objects due to gravity was Galileo Galilei.
• There is story that he dropped two iron balls of different masses from the top of the Leaning Tower of Pisa.
• Thereby, he demonstrated that gravity accelerates all masses at the same rate, provided that air resistance is negligibly small.
  By such an experiment you could calculate the acceleration due to gravity.
  using the equation:
  
• For example, Galileo’s assistant could have timed such a fall (from the top of the tower, which is 55m high) as three and a quarter swing of his 1-second pendulum.
  

⇒ Terminal speed

• Terminal speed, also known as terminal velocity, is the maximum speed that an object can reach as it falls through a fluid, such as air or water.
• At this point, the force of gravity pulling the object down is balanced by the frictional force of the fluid pushing the object up, and the object no longer accelerates.
• Terminal speed depends on several factors, including:
  1. Mass: Heavier objects tend to have a higher terminal speed.
  2. Shape: Objects with a more streamlined shape can reach a higher terminal speed.
  3. Size: Larger objects tend to have a higher terminal speed.
  4. Density: Objects with a higher density than the surrounding fluid tend to have a higher terminal speed.
  5. Fluid properties: The viscosity and density of the fluid affect terminal speed.
• Drag is the name given to resistive forces experienced by an object moving through a fluid such as air or water.
• Depends on:
  – Shape and size of the object
  – Velocity of the object
  – Density of the fluid
  – Viscosity of the fluid (thickness or “stickiness”)
  – Acts in the opposite direction of motion

Figure 6 At this instant the two balls fall at the same speed; the drag is the same on each but the blue ball continues to accelerate.

3. Projectile motion – or falling sideways

• Projectile motion is a fundamental concept in physics that describes the motion of an object that is thrown, launched, or projected into the air.
• It’s a two-dimensional motion, meaning the object moves in both horizontal and vertical directions.
• the key aspects of projectile motion:
  1. Trajectory: The path the object follows under the influence of gravity.
  2. Initial velocity (v_i): The velocity at which the object is launched.
  3. Angle of launch (θ): The angle at which the object is launched, relative to the horizontal.
  4. Gravity (g): The acceleration due to gravity, which pulls the object downwards.
  5. Time of flight (T): The total time the object is in the air.
  6. Range (R): The horizontal distance the object travels.
  7. Maximum height (H): The highest point the object reaches.
• According to Newton’s first Law of motion, unless an unbalanced force acts upon an item, it will continue to travel in a straight path at a constant speed or remain at rest.
• A resultant force applied on an object causes it to accelerate in the direction stated by Newton’s second law of motion.
• Force = mass * acceleration
• Projectile paths may be predicted using the equations of motion.
• A vector velocity may be resolved into horizontal and vertical components.

⇒ Falling sideways

• A diver may be seen leaping into the water in Figure 7.
• Every picture is captured 0.2 seconds apart.

  
  Figure 7 A man jumping off a diving board into the sea

• It is evident that the diver’s sideways displacement is consistent across all photo frames.
• This is because he is not being affected by any horizontal force, therefore he continues to go in that direction at a steady speed.
• However, his downward displacement gets larger with every image frame.
• This is because he is falling faster due to the acceleration caused by gravity.
• The diver’s ability to move both vertically and horizontally independently of one another is a crucial concept.
• Since the diver has no beginning velocity in a vertical direction, he will always reach the water at the same moment when he jumps horizontally from the diving board.
• But the quicker he sprints sideways, the farther his fall will take him from the board.
• In all of the calculations that follow, we shall assume that we may ignore the effects of air resistance.
• However, when something is moving very quickly.
• A golf ball for example
  • The effects of air resistance need to be taken into account.
  • The effect of air resistance on the path of a golf ball is to reduce its maximum height and to reduce the horizontal distance (range) that it travels.