Force and Motion notes

MDCAT Physics

This chapter explains how the position and motion of an object are described using displacement, velocity and acceleration. It also covers projectile and circular motion, Newton's laws, momentum, impulse, collisions, equilibrium and the limits of classical mechanics.

Displacement, Distance and Velocity

Distance is the total length of the path travelled by an object. It is a scalar quantity and has only magnitude. Displacement is the shortest directed distance from the initial position to the final position. It is a vector quantity.

For motion along the x-axis, displacement is calculated by Δx = xf − xi. Its sign shows direction. Average velocity is the net displacement divided by the total time, while average speed is the total distance divided by the total time.

Distance is a scalar quantity, while displacement is a vector quantity.
Distance is always greater than or equal to the magnitude of displacement.
Distance equals the magnitude of displacement when the object moves along a straight path without changing direction.
For a complete round trip, displacement is zero but distance is not zero.
Average velocity = total displacement / total time.
Average speed = total distance / total time.
Instantaneous velocity is the velocity at a particular instant and is given by v = dx/dt.
If xi = −4 m and xf = −8 m, then Δx = −8 − (−4) = −4 m and the magnitude of displacement is 4 m.

Acceleration and Equations of Motion

Acceleration is the rate of change of velocity with time. It is a vector quantity. Uniform acceleration means that velocity changes by equal amounts in equal intervals of time.

For motion with constant acceleration, the standard equations of motion are used. The value of g near the Earth's surface is commonly taken as 9.8 m/s2 or approximately 10 m/s2 in numerical questions.

Acceleration = change in velocity / time, a = (v − u)/t.
First equation: v = u + at.
Second equation: s = ut + 1/2 at2.
Third equation: v2 = u2 + 2as.
For uniform acceleration, s = (u + v)t/2.
The slope of a velocity-time graph gives acceleration.
A velocity-time graph parallel to the time axis has zero slope, so acceleration is zero.
An object moving at constant velocity has zero acceleration. Constant speed alone does not always mean zero acceleration, because direction may change.
Example: For a body falling from rest through 1.25 m, v2 = 0 + 2(10)(1.25), so v = 5 m/s just before reaching the ground.
A train with u = 20 m/s, a = 2 m/s2 and s = 100 m has v2 = 202 + 2(2)(100) = 800, so v is approximately 28 m/s.

Projectile Motion

Projectile motion is the motion of an object projected into the air and then allowed to move under gravity. Air resistance is ignored in the ideal model. The horizontal and vertical motions are treated separately.

The horizontal component of velocity remains constant because horizontal acceleration is zero. The vertical component changes because the object has a downward acceleration g. The path of an ideal projectile is a parabola.

For projection speed u at angle θ, horizontal velocity is ux = u cosθ.
Initial vertical velocity is uy = u sinθ.
Horizontal displacement is x = u cosθ × t.
Vertical displacement is y = u sinθ × t − 1/2 gt2.
At the highest point, vertical velocity is zero, but horizontal velocity is not zero.
At the highest point, acceleration is still g downward.
Time of flight for a projectile that lands at the same level is T = 2u sinθ/g.
Maximum height is H = u2 sin2θ/(2g).
Horizontal range for equal initial and final levels is R = u2 sin2θ/g.
A projectile has maximum range at 45° when air resistance is ignored.
A ballistic missile is an unpowered and unguided projectile after its launching phase. Rockets use thrust and may have variable acceleration because their mass changes as fuel is expelled.
If a projectile is fired at 60° with kinetic energy E, its horizontal kinetic energy at the highest point is E cos2 60° = E/4.

Circular and Angular Motion

An object moving in a circle may have constant speed but changing velocity because its direction continuously changes. Therefore, uniform circular motion is accelerated motion. The net acceleration is directed towards the centre of the circle.

The inward force required to maintain circular motion is called centripetal force. It is not a new type of force. It may be provided by tension, friction, gravity or electrostatic force, depending on the situation.

Centripetal acceleration is ac = v2/r = ω2r.
Centripetal force is Fc = mv2/r = mω2r.
Centripetal force always acts towards the centre of the circular path.
The net acceleration of a particle in circular motion is towards the centre.
For an electron moving around a nucleus in the classical model, the required centripetal force is provided by Coulomb force.
Angular displacement is measured in radians. One complete revolution equals 2π radians.
Angular velocity is the rate of change of angular displacement, ω = Δθ/Δt.
Angular velocity at any particular instant is called instantaneous angular velocity.
The relation between linear and angular speed is v = rω.
One revolution per second is one hertz. Angular speed is related to frequency by ω = 2πf.
For 3600 revolutions per minute, f = 3600/60 = 60 revolutions per second and ω = 2π(60) = 120π rad/s. In one second, the angular displacement is 120π radians. The angle for one revolution alone is 2π radians.

Newton's Laws of Motion

Newton's laws describe the relation between force and motion in inertial frames of reference. An inertial frame is one that is at rest or moving with constant velocity relative to another inertial frame.

A force can change the state of rest, uniform motion, direction or shape of an object. The net force is the vector sum of all forces acting on the body.

Newton's first law is also called the law of inertia.
If the net external force is zero, an object remains at rest or continues moving with constant velocity.
Newton's second law states that the net force is equal to the time rate of change of momentum: F = dp/dt.
For constant mass, Newton's second law becomes F = ma.
The SI unit of force is the newton: 1 N = 1 kg m/s2.
A 10 kg mass moving with acceleration 10 m/s2 has force F = ma = 100 N.
A body is in translational equilibrium when the net force on it is zero.
Equilibrium does not necessarily mean that the body is at rest. It may move with constant velocity.
Newton's third law states that for every action, there is an equal and opposite reaction.
The action and reaction forces act on different bodies, so they do not cancel each other on one free-body diagram.
Newton's laws are not directly valid in an accelerated, or non-inertial, frame without introducing suitable fictitious forces.
Classical Newtonian laws are not adequate for particles moving with velocities comparable to the velocity of light. Relativistic mechanics is required in that situation.

Angular Form of Newton's Second Law

Rotational motion has quantities corresponding to linear motion. Torque produces angular acceleration in the same way that force produces linear acceleration. The rotational form of Newton's second law connects torque with the moment of inertia and angular acceleration.

Torque is the turning effect of a force about an axis. Its value depends on the force, the perpendicular distance from the axis and the angle between the force and the lever arm.

Torque is τ = rF sinθ, where r is the distance from the axis to the point of application.
The SI unit of torque is N m.
Moment of inertia is the rotational equivalent of mass. It depends on mass distribution relative to the axis.
Angular acceleration is the rate of change of angular velocity, α = dω/dt.
Angular form of Newton's second law is τ = Iα.
A larger moment of inertia produces smaller angular acceleration for the same torque.
A force passing through the axis produces zero torque because its perpendicular distance is zero.
For a rigid body rotating about a fixed axis, rotational kinetic energy is K = 1/2 Iω2.

Linear Momentum and Impulse

Linear momentum is a measure of the quantity of motion of a body. It depends on both mass and velocity. Momentum is a vector, so its direction is the same as the direction of velocity.

Impulse is the effect of a force acting over a time interval. It equals the change in momentum. A large force acting for a short time can produce the same impulse as a smaller force acting for a longer time.

Linear momentum is p = mv.
The SI unit of momentum is kg m/s, which is also equivalent to N s.
The time rate of change of momentum is force: F = Δp/Δt or F = dp/dt.
Impulse J = FΔt when force is constant.
Impulse is equal to change in momentum: J = Δp = pf − pi.
A 10 N force acting for 10 s produces impulse J = 10 × 10 = 100 N s, equal to a momentum change of 100 kg m/s.
Momentum is conserved when the net external force on a system is zero.
Internal forces cannot change the total momentum of an isolated system.
Airbags and seat belts increase the time over which momentum changes, reducing the average force on a person.

Collisions and Conservation of Momentum

A collision is a short-duration interaction between two or more bodies during which large forces act. Momentum is conserved for an isolated system, whether or not kinetic energy is conserved.

In a collision, always choose a positive direction and use signs for velocities. The algebraic total momentum before collision equals the algebraic total momentum after collision.

For two bodies, conservation of momentum is m1u1 + m2u2 = m1v1 + m2v2.
An elastic collision conserves both total momentum and total kinetic energy.
An inelastic collision conserves momentum but does not conserve total kinetic energy.
In a perfectly inelastic collision, the bodies stick together after impact and move with a common velocity.
For a perfectly inelastic collision, common velocity V = (m1u1 + m2u2)/(m1 + m2).
Kinetic energy lost in an inelastic collision is converted into heat, sound, deformation or other forms of energy.
Momentum may be conserved even when kinetic energy is not conserved.
A collision with a wall changes the momentum of the object because its velocity changes direction.

Graphs, Uncertainty and Vector Checks

Physical quantities must be interpreted with their directions and units. Graphs provide a visual method for finding motion quantities. Measurement uncertainty describes the possible range within which the true value may lie.

For addition and subtraction, absolute uncertainties are combined. For multiplication and division, fractional or percentage uncertainties are combined.

The slope of a displacement-time graph gives velocity.
The slope of a velocity-time graph gives acceleration.
The area under a velocity-time graph gives displacement.
The area under an acceleration-time graph gives change in velocity.
For a result obtained by subtraction, total absolute uncertainty equals the sum of the absolute uncertainties of the two measurements.
A vector equation must be resolved into perpendicular components before applying scalar equations.
The magnitude of a vector is never negative, although a component or displacement may have a negative sign.
Always write the final numerical answer with its SI unit when a unit is required.

Key terms

Distance: The total length of the actual path travelled by an object.
Displacement: The directed change in position from the initial point to the final point.
Velocity: The rate of change of displacement with time.
Acceleration: The rate of change of velocity with time.
Projectile: An object moving through the air under the effect of gravity after projection.
Centripetal force: The inward force required to keep an object moving in a circular path.
Angular velocity: The rate of change of angular displacement.
Torque: The turning effect of a force about an axis.
Inertia: The tendency of a body to resist a change in its state of rest or motion.
Force: An interaction that can change the motion or shape of a body.
Momentum: The product of mass and velocity, p = mv.
Impulse: The product of force and time interval, equal to the change in momentum.
Equilibrium: The condition in which the net force on a body is zero.
Elastic collision: A collision in which both momentum and total kinetic energy are conserved.
Inelastic collision: A collision in which momentum is conserved but total kinetic energy is not conserved.
Inertial frame: A reference frame in which Newton's laws hold without fictitious forces.
Moment of inertia: The rotational equivalent of mass, determined by how mass is distributed about an axis.
Uncertainty: The estimated range of doubt associated with a measured physical quantity.

Test yourself on Force and Motion

Free Force and Motion MCQs with an explanation on every answer. No account needed.

Practise Force and Motion MCQs

More for Force and Motion in the MDCAT pack

A one-page revision sheet for this chapter
5 Force and Motion mnemonics
Chapter-wise Ratta Cards and a Quiz Builder for your own tests

See the MDCAT packAlready have it? Open Revision

Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

15 more Physics shortcuts are in the MDCAT pack. Already have it? See all shortcuts