High School Physics for JAMB/Motion

Introduction Motion is defined as the change in an object's position relative to a reference point over time. The branch of motion which deals with the description of how objects move without regard to forces causing their motion is called kinematics, while the one that deals with why they move, that is, what causes them to move as they do, is called dynamics. Types of Motion These are the major types of motions: Translational Motion This is a type of motion along a straight line (rectilinear) or curve (curvilinear), without rotating. For example, a car driving in a straight line, a drifting car along a curved racetrack, or an apple falling. It is also referred to as linear motion.none|thumb|250x250px|An example of a ball moving in a straight line (linear motion) Oscillatory Motion This is a type of motion that involves repeated back-and-forth (to-and-fro) motion around a mean, central position. This motion is periodic as it repeats itself at regular intervals. For example, a pendulum, a swing, or a guitar string. It is also referred to as vibratory motion. none|thumb|250x250px|An illustration of a simple pendulum Rotational Motion This is a type of motion that involves an object turning around a fixed axis. For example, a spinning top, a ceiling fan, or the Earth rotating. Spin This is a type of motion that involves the rotational movement of a rigid body around an internal axis that passes directly through its own center of mass.: How to differentiate between spin and rotational motion: Rotational motion is the umbrella term for any object turning around a fixed axis, while spin is a specific type of rotational motion where the axis goes directly through the object itself (an internal axis). For example, the Earth spins on its own axis, causing day and night (not its orbit around the Sun), or a ball rotating on a player's finger. none|thumb|500x250px|A sphere rotating (spinning) about an axis Random Motion This is a type of motion with no specific path or pattern. The particles collide with one another and with the walls of their containers, if they are kept in containers. The Brownian motion is an example of random motion involving moving gas molecules and a flying bee.none|thumb|250x250px|An illustration of the random motion of a particle. Relative Motion Motion is defined as a change in position. However, it is impossible to determine this change without a reference point or a frame of reference. This means that any measurement used to assess the motion of an object must be made in relation to these reference points or frames. Relative motion refers to the motion of one object in relation to another. All objects moving on the Earth's surface exhibit relative motion with respect to the Earth, which is generally assumed to be stationary. For instance, with respect to the Earth, if a boy is standing on top of a moving bus, his speed is the same as the speed of the bus. If the boy runs in the same direction as the bus, his overall speed is the sum of the bus's speed and his running speed. Conversely, if he runs in the opposite direction, his speed is the difference between the bus's speed and his running speed. none|thumb|400x400px|An illustration of the given example Causes of Motion The primary cause of motion is a Force. A force is simply defined as a push or a pull exerted on an object. Sir Isaac Newton established that an object will remain at rest or continue to move at a constant velocity unless acted upon by an external, unbalanced force. Thus, force is the agent that changes or tends to change the state of rest or uniform motion of a body. Types of Force Forces are broadly classified into two main categories depending on whether physical contact is required between the interacting bodies. Contact Forces These are forces that require physical contact between two objects to be experienced. They occur when two interacting objects are touching each other. Frictional Force: The force that opposes the relative motion of two surfaces in contact. Tensional Force: The force transmitted through a string, rope, or wire when it is pulled tight by forces acting from opposite ends. Normal Reaction: The upward force exerted by a surface on an object resting on it, perpendicular to the surface. Upthrust (Buoyancy): The upward force exerted by a fluid on an object completely or partially immersed in it. Force Fields These are forces whose sources do not require physical contact with the body to which they are applied. They act through empty space over a distance. Gravitational Force: The pull of the Earth (or any massive body) on objects. This is what gives objects weight and causes them to fall. Electrostatic Force: The force of attraction or repulsion between electrically charged bodies. Magnetic Force: The force of attraction or repulsion between magnetic poles. Linear Motion Linear motion (or rectilinear motion) is the motion of a body along a straight line. Speed, Velocity, and Acceleration Speed: This is the rate of change of distance with time. It is a scalar quantity (has only magnitude). Speed = Distance / Time (). The SI unit is meters per second or . Velocity: This is the rate of change of displacement with time. It is a vector quantity (has both magnitude and direction). Velocity = Displacement / Time (). The SI unit is or . Acceleration: This is the rate of change of velocity with time ( or ). It is a vector quantity. When an object slows down, it experiences negative acceleration, commonly called deceleration or retardation. The SI unit is or . Equations of Uniformly Accelerated Motion When an object moves with a constant acceleration in a straight line, its motion can be described using the following equations of kinematics: Where: = initial velocity () = final velocity () = constant acceleration () = time taken (s) = distance/displacement covered (m) Motion Under Gravity When an object is thrown vertically upwards or dropped from a height, it moves under the influence of the Earth's gravity. The acceleration due to gravity, denoted as , is approximately or (as often used in JAMB). For an object falling downwards, acceleration is positive (). For an object thrown upwards against gravity, acceleration is negative (). Note: The linear motion equations are modified by replacing with , and with height, . Distance-Time Graph and Velocity-Time Graph Graphical representations help us analyze motion easily: Distance-Time Graph: A plot of distance on the vertical axis against time on the horizontal axis. The slope (gradient) of a distance-time graph represents the object's speed. Velocity-Time Graph: A plot of velocity on the vertical axis against time on the horizontal axis. The slope (gradient) represents the acceleration. The area under the velocity-time graph represents the total distance (or displacement) covered by the body. Instantaneous Velocity and Acceleration Instantaneous Velocity: This is the velocity of an object at a specific, exact moment in time. On a displacement-time graph, it is calculated by finding the slope of the tangent to the curve at that specific instant. Instantaneous Acceleration: This is the acceleration of a body at a specific instant in time. It is found by determining the slope of the tangent to the curve on a velocity-time graph at that specific point. Projectiles A projectile can be defined as any object thrown into space upon which the only acting force is gravity. The path followed by a projectile is called its trajectory, which takes the shape of a parabola. Calculation of Range, Maximum Height, and Time of Flight If a projectile is launched from the ground with an initial velocity at an angle to the horizontal: Time of Flight (): The total time the projectile spends in the air before hitting the ground. Time to Max Height (): This is the time taken for a projectile to attain maximum height. Maximum Height (): The highest vertical distance reached by the projectile. At this point, the vertical component of its velocity is zero. Horizontal Range (): The total horizontal distance traveled from the point of projection to the point where it hits the landing plane. Note: Maximum range occurs when the angle of projection is 45°. Derivation of the projectile formulas Using the equations of kinematics and replacing the with , and the with , we get: Resolving to the vertical, we get: = Time to max height () At maximum height, as the velocity slowly decreases to the peak (max height), to which it becomes zero, and then increases until it touches the ground. Then, becomes as it goes againt gravity Time of flight () Time of flight is twice the time taken to reach the maximum height, . This is because it follows the same path down as illustration in the diagram below: none|This time taken reach max height is same as the time taken to land.|thumb|250x250px Maximum Height () Using the formula, , we can make the subject of the formula. ( because of the same reason above) Horizontal Range () We can derive the horizontal range by multiplying the horizontal component of velocity by the total time taken for the flight. Using the trigonometric identity where, , we can simplify further, yielding: Applications of Projectile Motion Projectile motion concepts are highly applicable in various real-world scenarios: Sports: Kicking a football, serving a volleyball, throwing a javelin, or hitting a golf ball. Military: Firing artillery shells, launching non-guided missiles, and dropping bombs from an aircraft. Aviation: Dropping relief packages from an airplane to a specific target on the ground. Newton's Laws of Motion Inertia, Mass, and Force Newton's First Law (Law of Inertia): States that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. Inertia is the tendency of a body to resist a change in its state of rest or uniform motion. Mass is the quantitative measure of a body's inertia. The heavier the object, the greater its inertia. Force is what is required to overcome this inertia. Relationship Between Mass and Acceleration Newton's Second Law: States that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force. Mathematically, this translates to the famous equation: Where is Force, is mass, and is acceleration. This shows that for a constant mass, force is directly proportional to acceleration. Impulse and Momentum Momentum (): The product of an object's mass and its velocity. . Impulse (): The product of a force and the time interval over which it acts. Impulse is equal to the change in momentum. Both impulse and momentum share the same SI unit: Newton-seconds (Ns) or kilogram-meters per second (kg m/s). Force-Time Graph When a variable force is plotted against time, the area under the force-time graph is equal to the Impulse (which is also the change in momentum) of the object. Conservation of Linear Momentum Note: Coefficient of restitution is not covered in this section. The Principle of Conservation of Linear Momentum states that in a closed system of colliding bodies, the total momentum before collision is equal to the total momentum after collision, provided no external resultant force acts on the system. This principle is widely applied in understanding the recoil of a gun and rocket propulsion. Motion in a Circle Angular Velocity and Angular Acceleration Angular Velocity (): The rate of change of angular displacement () with respect to time. It is measured in radians per second (rad/s). Linear velocity () is related to angular velocity by , where is the radius of the circle. Angular Acceleration (): The rate of change of angular velocity with time. Measured in . Centripetal and Centrifugal Forces Centripetal Force: This is the inward force required to keep an object moving in a circular path. It is always directed towards the center of the circle. Centrifugal Force: This is an apparent (or fictitious) outward force experienced by an object moving in a circle, equal and opposite to the centripetal force. thumb|An illustration of the direction of the centrifugal and centripetal forces|none|250x250px Applications Banking of roads and railway tracks to prevent vehicles from skidding off at curves. The centrifuge machine is used in laboratories to separate particles of different densities (like blood components). Satellites orbiting the Earth and planets orbiting the Sun. Simple Harmonic Motion (S.H.M) Definition and Explanation of S.H.M Simple Harmonic Motion is a special type of periodic, oscillatory motion where the restoring force (or acceleration) is directly proportional to the displacement from an equilibrium position and is always directed towards that equilibrium position. Mathematically: The negative sign indicates that acceleration is always directed towards the central mean position, opposite to the displacement (). Examples of Systems that Execute S.H.M A simple pendulum swinging through a small angle. A mass suspended on an elastic spring is bobbing up and down. Liquid oscillating in a U-tube. The prong of a sounding tuning fork. Period, Frequency, and Amplitude of S.H.M Amplitude (): The maximum displacement of the oscillating particle from the mean position. Period (): The time taken to complete one full cycle or oscillation. Frequency (): The number of complete oscillations made in one second. Measured in Hertz (Hz). Relationship: For a simple pendulum: For a loaded spring: Velocity and Acceleration of S.H.M Velocity (): The velocity of a particle in S.H.M is maximum at the equilibrium (mean) position and zero at the extreme positions (amplitude). Acceleration (): Acceleration is maximum at the extreme positions and zero at the mean position. Simple Treatment of Energy Change in S.H.M In S.H.M, there is a continuous conversion between Kinetic Energy (K.E) and Potential Energy (P.E), but the total mechanical energy remains constant. At the mean position: K.E is maximum, P.E is zero. At the extreme positions: P.E is maximum, K.E is zero. Total Energy = . Forced Vibration and Resonance Forced Vibration: This occurs when a system is continuously driven by an external periodic force to vibrate at a frequency different from its natural frequency. Resonance: This is a phenomenon that occurs when the frequency of the externally applied periodic force matches the natural frequency of the oscillating body. When this happens, the system absorbs maximum energy and vibrates with its maximum amplitude. Examples: Tuning a radio station, acoustic resonance in musical instruments, or a wine glass shattering when a singer hits a specific high pitch. Past Questions Quiz Types of Motion JAMB Past Questions JAMB 1980 The motion of the moving skin of a talking drum can rightly be described as: A) translational B) random C) rotational D) oscillatory E) transitory D) oscillatory The skin of a talking drum vibrates back and forth about a fixed central position when struck. This to-and-fro periodic motion is the classic definition of oscillatory (or vibratory) motion. JAMB 2014 Which type of motion do the wheels of a moving car undergo? A) Vibratory and translational motion B) Random and translational motion C) Rotational and oscillatory motion D) Translational and rotational motion D) Translational and rotational motion As a car moves, its wheels turn around an axis (rotational motion) while simultaneously moving forward along the road (translational motion). JAMB 1995 When a ball rolls on a smooth level ground, the motion of its center is: A) translational B) oscillatory C) random D) rotational A) translational While the ball as a whole experiences both rotational and translational motion, the specific point at its center only moves in a straight line relative to the ground. Therefore, the center undergoes pure translational (rectilinear) motion. Linear Motion & Kinematics I JAMB Past Questions JAMB 2009 A car accelerates uniformly from rest at . How far will it travel in the fifth complete second? A) 100 m B) 50 m C) 32 m D) 18 m D) 18 m The distance traveled in the second is given by the formula: .Given , , and : . JAMB 1994 A motor vehicle is brought to rest from a speed of in 20 seconds. Calculate the retardation. A) B) C) D) A) Using the equation of motion: .The final velocity (since it is brought to rest), , . . Retardation is negative acceleration, so its magnitude is . JAMB 2017 Which is the incorrect formula for a body accelerating uniformly? A) B) C) D) B) The correct third equation of motion is . Option B misses the square on the initial velocity (). Linear Motion & Kinematics II JAMB Past Questions JAMB 1997 The distance (in meters) travelled by a particle in time (in seconds) is described by the equation . Find the average speed of the particle between the time interval and . A) 60 m/s B) 72 m/s C) 84 m/s D) 108 m/s C) 84 m/s First, calculate the distance at both time intervals:At : At : Average speed = Average speed = . OR, alternatively: Speed Differentiating the formula, we get: At : At : Average speed JAMB 1981 Two particles X and Y starting from rest cover the same distance. The acceleration of X is twice that of Y. The ratio of the time taken by X to that taken by Y is: A) B) C) D) E) C) From the equation , since , we have , which means .For particle X: and for Y: .Ratio . Since , the ratio is . JAMB 1983 A train has an initial velocity of and an acceleration of . Its velocity after 10 seconds is: A) 2 m/s B) 4 m/s C) 8 m/s D) 12 m/s E) 16 m/s B) 4 m/s Using the first equation of motion: . Motion Under Gravity JAMB Past Questions JAMB 1978 A palm fruit dropped to the ground from the top of a tree 45m tall. How long does it take to reach the ground? () A) 9s B) 4.5s C) 6s D) 7.5s E) 3s E) 3s For a freely falling body, initial velocity . Using : . JAMB 1981 A body falls freely under gravity () from a height of 40m on to the top of a platform 0.8m above the ground. Its velocity on reaching the platform is: A) 784 m/s B) 80 m/s C) 78.4 m/s D) 39.2 m/s E) 27.7 m/s E) 27.7 m/s The actual distance fallen by the body is .Using (where ):. JAMB 1991 A small metal ball is thrown vertically upwards from the top of a tower with an initial velocity of . If the ball took a total of 6s to reach ground level, determine the height of the tower. () A) 60m B) 80m C) 100m D) 120m A) 60m Taking the top of the tower as the reference point, the final displacement is negative because it lands below the projection point.Using :.The height of the tower is 60m. Projectile Motion JAMB Past Questions JAMB 1994 An object is projected with a velocity of at an angle of 30° to the horizontal. The maximum height reached is: A) 20m B) 80m C) 160m D) 320m B) 80m Maximum height .. JAMB 1997 Two bodies X and Y are projected on the same horizontal plane, with the same initial speed but at angles 30° and 60° respectively to the horizontal. Neglecting air resistance, the ratio of the range of X to that of Y is: A) 1:1 B) 1:2 C) √3:1 D) √3 A) 1:1 The horizontal range formula is . Complementary angles (angles that sum up to 90°, like 30° and 60°) always yield the exact same horizontal range provided their initial speeds are identical. Thus, the ratio is 1:1. JAMB 2006 A shooter wants to fire a bullet in such a way that its horizontal range is equal to three times its maximum height. At what angle should he fire the bullet to achieve this? A) 68° B) 53° C) 30° D) 45° B) 53° Range .Cancel out from both sides: .Using the identity :.Divide both sides by : .. Newton's Laws & Inertia JAMB Past Questions JAMB 1984 Which of the following is a correct explanation of the inertia of a body? A) Ability to overcome the earth's gravity B) Reluctance to stop moving C) Readiness to start moving D) Reluctance to start moving and its reluctance to stop moving once it has begun to move E) Reluctance to overcome the earth's gravity D) Reluctance to start moving and its reluctance to stop moving once it has begun to move Inertia is the inherent property of matter that resists any change to its state of rest or uniform motion in a straight line, as defined by Newton's First Law of Motion. JAMB 1992 A boy sits in a train moving with uniform speed on a straight track. If from his outstretched palm he gently tosses his coin upwards, the coin will fall: A) in front of his palm B) behind his palm C) beside his palm D) into his palm D) into his palm Due to inertia of motion, the coin retains the horizontal velocity of the train even when tossed upwards. Since the train is moving at a uniform speed, the coin travels horizontally at the same speed as the boy's hand and will fall right back into his palm. JAMB 2004 I. Jet propelled aircraftII. Rocket propulsionIII. The recoil of a gunIV. A person walkingWhich of the above is based on Newton's third law of motion? A) I and II only B) I, II, III, and IV C) I and III only D) I, II and III only B) I, II, III, and IV Newton's Third Law states that for every action, there is an equal and opposite reaction. This principle applies to a jet/rocket pushing gases backward to go forward, a gun recoiling backward when a bullet shoots forward, and a person pushing the ground backward to walk forward. Momentum, Impulse & Work JAMB Past Questions JAMB 1978 A machine gun with a mass of 5kg fires a 50g bullet at a speed of . The recoil speed of the machine gun is: A) 0.5 m/s B) 1.5 m/s C) 1.0 m/s D) 2.0 m/s E) 4.0 m/s C) 1.0 m/s According to the principle of conservation of linear momentum, the momentum of the bullet equals the recoil momentum of the gun. . JAMB 1998 A ball of mass 0.15kg is kicked against a rigid vertical wall with a horizontal velocity of . If it rebounded with a horizontal velocity of , calculate the impulse of the ball on the wall. A) 3.0 Ns B) 4.5 Ns C) 7.5 Ns D) 12.0 Ns D) 12.0 Ns Impulse is the change in momentum: . Since it rebounds, the final velocity is in the opposite direction. Let and .. The magnitude of the impulse is . Required Question The area under a force-distance graph represents: A) acceleration B) velocity C) power D) work E) momentum D) work The area under a Force-Distance (or Force-Displacement) graph is the product of Force and Distance, which is the mathematical definition of Work Done (or Energy transferred). (Note: Contrast this with the area under a Force-Time graph, which represents Impulse or Change in Momentum). Circular Motion I JAMB Past Questions JAMB 1978 The force required to make an object of mass , travelling with velocity , turn in a circle of radius is: A) B) C) D) E) A) The inward force that keeps an object moving in a circular path is called the centripetal force. It is mathematically given by the formula . JAMB 1984 A particle moves in a circular orbit of radius 0.02m. If the speed of the particle is , calculate its frequency in cycles per second. (Take ) A) 2.0 Hz B) 7.0 Hz C) 8.8 Hz D) 14.0 Hz E) 17.6 Hz B) 7.0 Hz The relationship between linear speed and angular velocity is .. JAMB 1994 Which of the following is TRUE of a particle moving in a horizontal circle with constant angular velocity? A) The kinetic energy is constant but the linear momentum varies B) The linear momentum is constant but the energy varies C) Both energy and linear momentum are constant D) The speed and the linear velocity are both constant A) The kinetic energy is constant but the linear momentum varies In uniform circular motion, the speed is constant, meaning the kinetic energy () remains constant. However, because the direction of motion is constantly changing, the velocity changes continuously. Since momentum is a vector quantity (), the changing direction means the linear momentum is also constantly varying. Circular Motion II JAMB Past Questions JAMB 2007 A car of mass 1500 kg goes round a circular curve of radius 50m at a speed of . The magnitude of the centripetal force on the car is? A) B) C) D) A) Centripetal force . JAMB 2000 I. Its velocity is constantII. No work is done on the bodyIII. It has constant acceleration directed away from the centreIV. The centripetal force is directed towards the centreWhich combination of the above is true of a body moving with constant speed in a circular track? A) II and IV B) I and IV C) II and III D) I and III A) II and IV Statement I is false because velocity changes as direction changes. Statement III is false because acceleration is directed towards the center. Statement II is true because the centripetal force is perpendicular to the displacement, so . Statement IV is true by definition of centripetal force. JAMB 2008 A motorcycle of mass 100kg moves round in a circle of radius 10m with a velocity of . Find the coefficient of friction between the road and the tyres. () A) 25.00 B) 2.50 C) 0.50 D) 0.25 D) 0.25 For a vehicle moving in a circular path, the necessary centripetal force is provided by the frictional force. . Simple Harmonic Motion (SHM) JAMB Past Questions JAMB 1997 Which of the following with respect to a body performing simple harmonic motion are in phase? A) displacement and velocity of the body B) displacement and force on the body C) velocity and acceleration of the body D) force acting on the body and the acceleration D) force acting on the body and the acceleration According to Newton's Second Law (), force and acceleration are always directly proportional and act in the exact same direction. Therefore, they are always in phase with one another. JAMB 2021 The motion of a body is simple harmonic if the? A) acceleration is always directed towards a fixed point and directly proportional to displacement B) path of motion is a straight line C) acceleration is proportional to the square of the distance from a fixed point D) acceleration is constant and directed towards a fixed point A) acceleration is always directed towards a fixed point and directly proportional to displacement The strict condition for Simple Harmonic Motion (SHM) is that the acceleration must be directly proportional to its displacement from an equilibrium point, and it is always directed towards that fixed point (). JAMB 2019 A body moves in SHM between two points 20m apart on a straight line. If the angular speed of the body is , calculate its speed when it is 6m from the center of the motion. A) 20 m/s B) 30 m/s C) 40 m/s D) 50 m/s C) 40 m/s The distance between the two extreme points is 20m, meaning the amplitude . Using the velocity formula for SHM: . Graphs of Motion JAMB Past Questions JAMB 1984 The distance travelled by a particle starting from rest is plotted against the square of the time elapsed from the commencement of motion. The resulting graph is linear, the slope of this graph is a measure of: A) initial displacement B) initial velocity C) acceleration D) half the acceleration E) half the initial velocity D) half the acceleration According to the second equation of motion, . Since it starts from rest, , so the equation becomes . If you plot distance () on the y-axis against the square of time () on the x-axis, the equation represents a straight line () where the slope . Therefore, the slope measures half the acceleration. JAMB 2021 The slope of the straight line displacement-time graph indicates? A) distance traveled B) uniform velocity C) uniform acceleration D) uniform speed B) uniform velocity On a displacement-time graph, the slope (or gradient) is defined as the change in displacement over the change in time (), which is the exact definition of velocity. Because the line is straight, the slope is constant, indicating uniform velocity. JAMB 1989 Two points on a velocity time-graph have coordinates (5s, ) and (20s, ). Calculate the mean acceleration between the two points. A) B) C) D) A) Mean acceleration is the slope of the velocity-time graph..