Definitions [17]
Define Distance.
The length of path travelled by a body in certain interval of time is called distance.
Define Displacement.
Displacement of an object between two points is the shortest distance between these two points.
“It is the unique path that can take the body from its initial to the final position.”
The displacement of a moving body is defined as the change in its position along a particular direction
- Displacement is the shortest distance in a straight line between the starting point and the final point, along with a direction. It is a vector quantity, meaning it includes both magnitude (distance) and direction.
- Example: In Ranjit’s case, the displacement is the straight line AD from his house to the school.
- Distance is the total length of the path travelled by a moving object, regardless of the direction. It is a scalar quantity, which means it only has magnitude (size), not direction.
- Example: If Ranjit walks from point A to B to C to D, the distance is the total of AB + BC + CD.
"dy/dx is called the derivative of y with respect to x (which is the rate of change of y with respect to change in x) and the process of finding the derivative is called differentiation."
is called the indefinite (without any limits on x) integral of f(x).
The representation \[\int_{x=a}^{x=b}\] f(x)dx is called the definite integral of f(x) from x = a to x = b.
The acceleration acting on an object undergoing uniform circular motion, which always acts on the object along the radius towards the centre of the circular path, is called centripetal acceleration.
The time taken by the projectile to travel from the point of projection to the maximum height is called the time of ascent (tA).
The total maximum horizontal distance travelled by a projectile from the point of projection to the point where it hits the ground is called the horizontal range (R).
The maximum vertical height reached by the projectile — i.e., the distance travelled along the vertical (y) direction up to the highest point — is called the maximum height (H).
An object in flight after being thrown with some velocity that follows a curved path under the action of gravity is called a projectile.
OR
A body in free fall which is subjected to the force of gravity and air resistance only — which refers to the motion of bodies flung into the air — is called a projectile.
The total time for which the projectile remains in the air — from the moment it is projected to the moment it returns to the same level — is called the time of flight (T).
The time taken by the projectile to travel from the maximum height back to the ground is called the time of descent (tD).
The path followed by a projectile is called its trajectory.
The direction of projection with respect to the horizon which determines the shape of trajectory (vertical → vertical, oblique → parabolic, horizontal → half parabolic) is called the projection angle.
The magnitude of projection velocity — which, with a fixed projection angle, shows the length of trajectory or range — is called the projection speed.
Formulae [1]
| Quantity | Formula |
|---|---|
| Position after time t | x = (u cos θ)t, y = (u sin θ)t − \[\frac {1}{2}\]gt2 |
| Equation of trajectory | y = x tan θ − \[\frac {g}{2u^2 cos^2 θ}\] ⋅ x2 |
| Maximum height | H = \[\frac {u^2 sin^2 θ}{2g}\] |
| Time of flight | T = \[\frac {2u sin θ}{g}\] |
| Horizontal range | R = \[\frac {u^2 sin 2θ}{g}\] |
| Maximum range | Rmax = \[\frac {u^2}{g}\] at θ = 45° |
| Velocity after time ttt | vx = u cos θ, vy = u sin θ − gt |
| Speed | v = \[\sqrt {v_x^2+v_y^2}\] |
Key Points
- Horizontal range is maximum at 45° and reduces for any other angle of projection.
- A projectile has two simultaneous independent motions — constant horizontal and gravity-driven vertical.
- The path is a symmetric parabola — equal time up and down, equal speed at the same height.
