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Motion and force: Class 11 Physics
Displacement, velocity and acceleration, equations of uniformly accelerated motion, free fall, projectile motion, Newton's laws, momentum and impulse, conservation of momentum, collisions, and friction.
Key points
Distance is the length of the path travelled and is a scalar. Displacement is the straight line change in position and is a vector. Speed is distance divided by time. Velocity is displacement divided by time, in m/s. Acceleration is the rate of change of velocity, a = (v − u)/t, in m/s². In a velocity time graph the slope gives acceleration and the area under the graph gives displacement.
Equations of motion for constant acceleration (initial velocity u, final velocity v, time t, displacement s):
- v = u + at
- s = ut + ½at²
- v² = u² + 2as
- s = ½(u + v)t
Free fall is motion under gravity alone. Near the Earth's surface all bodies fall with the same acceleration g = 9.8 m/s² downwards if air resistance is ignored. For a body thrown upwards, the velocity is zero at the highest point, the time of ascent equals the time of descent, and the body returns with the speed it started with.
Projectile motion. A projectile is thrown with speed u at angle θ above the horizontal. Its two motions are independent:
- Horizontal: constant velocity ux = u cos θ, because no horizontal force acts.
- Vertical: constant acceleration g downwards, with uy = u sin θ.
- Time of flight T = 2u sin θ / g.
- Maximum height H = u² sin²θ / 2g.
- Range R = u² sin 2θ / g. The range is greatest at θ = 45°, and angles θ and 90° − θ give the same range.
- The path is a parabola.
Newton's first law (law of inertia): a body stays at rest or moves with constant velocity in a straight line unless an unbalanced force acts on it. Inertia is the property of a body to resist a change in its state of motion, and mass is its measure. Newton's second law: the acceleration of a body is directly proportional to the net force and inversely proportional to its mass, F = ma. One newton is the force that gives a mass of 1 kg an acceleration of 1 m/s².
Newton's third law: to every action there is an equal and opposite reaction. The two forces act on different bodies, so they do not cancel. A rocket moves forward because the gas it pushes backward pushes the rocket forward. A person walks because the ground pushes forward on the foot when the foot pushes backward on the ground.
Linear momentum is p = mv, a vector in kg m/s. Newton's second law in general form is F = Δp/Δt, so force equals the rate of change of momentum. Impulse is the product of force and the time it acts, J = FΔt = Δp, with the unit N s. A cricket fielder draws his hands back while catching: this increases the time of the stop, so the force on the hands is smaller.
The law of conservation of momentum: when no external force acts on a system, its total momentum stays constant. For two bodies, m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. It follows from Newton's second and third laws. It explains the recoil of a gun: the bullet gains forward momentum and the gun gains an equal backward momentum.
Collisions:
- Elastic collision: both momentum and kinetic energy are conserved, as with collisions between gas molecules.
- Inelastic collision: momentum is conserved but some kinetic energy changes into heat and sound.
- Perfectly inelastic collision: the bodies stick together and move with a common velocity, v = (m₁u₁ + m₂u₂)/(m₁ + m₂).
- In a one dimensional elastic collision, the relative speed of approach equals the relative speed of separation.
Friction is the force that opposes relative motion between two surfaces in contact:
- Static friction adjusts itself up to a maximum value, fs(max) = μs R.
- Kinetic (sliding) friction acts when the surfaces slide, fk = μk R, and is usually less than the maximum static friction.
- R is the normal reaction, and μ is the coefficient of friction, which has no unit.
- Friction does not depend on the area of contact, to a good approximation.
- On a horizontal surface R = mg. On a slope of angle θ, R = mg cos θ.
Study tip: Write the known quantities with signs and units first, and then pick the equation that contains exactly one unknown. For projectile questions, treat horizontal and vertical motion separately. For collisions, write the momentum equation before the energy equation.
Definitions and flashcards
Say the meaning first, then open the card to check.
Displacement
The shortest distance from the initial to the final position, with direction.
Acceleration
The rate of change of velocity.
Inertia
The property of a body that resists a change in its motion.
Momentum
The product of mass and velocity.
Impulse
The product of force and the time of action, equal to the change in momentum.
Projectile
A body thrown into the air that moves under gravity alone.
Elastic collision
A collision in which kinetic energy is conserved.
Coefficient of friction
The ratio of the friction force to the normal reaction.
Free fall
Motion in which gravity is the only force acting.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| First equation | v = u + at | Constant acceleration. |
| Second equation | s = ut + ½at² | Displacement after time t. |
| Third equation | v² = u² + 2as | Used when time is not given. |
| Newton's second law | F = ma | Net force on mass m. |
| Momentum | p = mv | Unit kg m/s. |
| Impulse | FΔt = mv − mu | Equals the change in momentum. |
| Range of projectile | R = u² sin 2θ / g | On level ground. |
| Friction | f = μR | R is the normal reaction. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
State Newton's second law and define the newton. (2 marks)
The acceleration of a body is directly proportional to the net force acting on it and inversely proportional to its mass, F = ma. One newton is the force that produces an acceleration of 1 m/s² in a body of mass 1 kg.
Why does the time of flight of a projectile depend only on the vertical motion? (3 marks)
The vertical and horizontal motions are independent. Gravity acts only vertically, so the horizontal velocity stays constant and does not affect how long the body stays in the air. The body lands when the vertical displacement returns to zero, which gives T = 2u sin θ / g.
State the law of conservation of momentum and give one example. (3 marks)
If no external force acts on a system, the total momentum of the system remains constant. When a gun fires a bullet, the total momentum before firing is zero. After firing, the bullet has forward momentum and the gun has equal backward momentum, so the total remains zero.
Explain why a passenger moves forward when a bus stops suddenly. (2 marks)
The passenger has inertia. When the bus stops, the body tends to remain in its state of motion, so it continues forward while the feet are stopped by friction with the floor.
Distinguish between elastic and inelastic collisions. (3 marks)
In an elastic collision both momentum and kinetic energy are conserved, and the bodies separate. In an inelastic collision only momentum is conserved. Part of the kinetic energy changes into heat, sound or deformation. In a perfectly inelastic collision the bodies stick together.
Define impulse and show that it equals the change in momentum. (5 marks)
Impulse is the product of the average force and the time for which it acts, J = FΔt. From Newton's second law, F = ma = m(v − u)/Δt. Multiplying both sides by Δt gives FΔt = mv − mu. So impulse equals the final momentum minus the initial momentum. Its unit is the newton second (N s), which is the same as kg m/s. A large force acting for a short time, as in a bat hitting a ball, can produce the same impulse as a small force acting for a long time.
Numericals with worked steps
A car moving at 10 m/s accelerates uniformly at 2 m/s² for 5 s. Find the final velocity and the distance covered.
- v = u + at = 10 + 2 × 5 = 20 m/s.
- s = ut + ½at² = 10 × 5 + ½ × 2 × 25 = 50 + 25 = 75 m.
- Check: s = ½(u + v)t = ½ × 30 × 5 = 75 m.
Answer: v = 20 m/s and s = 75 m.
A ball is thrown at 20 m/s at 30° above the horizontal. Find the time of flight, the maximum height and the range. Take g = 9.8 m/s².
- T = 2u sin θ / g = 2 × 20 × 0.5 / 9.8 = 20/9.8 = 2.04 s.
- H = u² sin²θ / 2g = 400 × 0.25 / 19.6 = 100/19.6 = 5.10 m.
- R = u² sin 2θ / g = 400 × 0.866 / 9.8 = 346.4/9.8 = 35.3 m.
Answer: T = 2.04 s, H = 5.10 m, R = 35.3 m.
A 1000 kg car moving at 20 m/s hits a wall and stops in 0.1 s. Find the average force on the car.
- Change in momentum = 0 − 1000 × 20 = −20000 kg m/s.
- F = Δp/Δt = −20000/0.1 = −2 × 10⁵ N.
- The negative sign shows the force opposes the motion.
Answer: 2 × 10⁵ N (magnitude).
A 2 kg trolley moving at 3 m/s collides with a 1 kg trolley at rest and they stick together. Find the common velocity and the loss of kinetic energy.
- Momentum before = 2 × 3 + 1 × 0 = 6 kg m/s.
- After: (2 + 1)v = 6, so v = 2 m/s.
- KE before = ½ × 2 × 9 = 9 J.
- KE after = ½ × 3 × 4 = 6 J.
- Loss = 9 − 6 = 3 J.
Answer: v = 2 m/s and the loss of kinetic energy is 3 J.
A 10 kg block on a rough floor (μk = 0.3) is pulled by a horizontal force of 50 N. Find its acceleration. Take g = 9.8 m/s².
- Normal reaction R = mg = 10 × 9.8 = 98 N.
- Friction f = μk R = 0.3 × 98 = 29.4 N.
- Net force = 50 − 29.4 = 20.6 N.
- a = F/m = 20.6/10 = 2.06 m/s².
Answer: 2.06 m/s².
Common mistakes
- Using the equations of motion when acceleration is not constant.
- Forgetting sign conventions: if upwards is positive, g is taken as −9.8 m/s² for a body thrown upwards.
- Thinking that action and reaction cancel. They act on different bodies, so they never cancel.
- Applying conservation of kinetic energy to every collision. Only momentum is conserved in all collisions.
- Using the weight mg as the normal reaction on an incline. The normal reaction there is mg cos θ.