Equations of Uniformly Accelerated Motion

Equation:

Where:

• $u$ = initial velocity
• $v$ = final velocity
• $a$ = acceleration
• $t$ = time taken

Proof:

From the definition of acceleration, we know that:

Multiplying both sides by time $t$ to eliminate $t$ from the denominator:

Adding $u$ to both sides:

Therefore, we get the equation:

Equation:

Where:

• $S$ = distance travelled
• $u$ = initial velocity
• $t$ = time taken
• $a$ = acceleration

Proof:

We begin by considering that the object moves with an initial velocity $u$ to a final velocity $v$ in time $t$.

The distance $S$ travelled by the object can be expressed using the formula for average velocity multiplied by time:

The average velocity of the object is the mean of the initial and final velocities:

Thus, the distance travelled becomes:

Now, substitute the first equation of motion, $v = u + at$, into the equation:

Simplifying the expression:

Finally:

Equation:

Where:

• $v$ = final velocity
• $u$ = initial velocity
• $a$ = acceleration
• $s$ = displacement

Proof:

We start with the first equation of motion:

Squaring both sides:

Expanding the square on the right-hand side:

From the second equation of motion, we know that:

Multiplying the entire equation by 2:

Substitute this into the previous equation:

Since $ut + \tfrac{1}{2}at^2 = s$, we have:

• $v = u + at$
• $S = ut + \frac{1}{2}at^2$
• $v^2 = u^2 + 2as$

Motion under gravity refers to the motion of a body when it is subjected only to the gravitational force of the Earth. The constant acceleration due to gravity is denoted by $g$, and its approximate value is:

The concept of gravitational force:

1. When two objects of different masses fall from the same height in air, the heavier one may appear to fall faster due to less influence from air resistance.
2. In a vacuum (where there is no air resistance), all objects fall at the same rate regardless of their mass.

Consider a body falling freely from a certain height $h$ and reaches the ground in time $t$. The following assumptions are made:

• Initial velocity, $u = 0$
• Acceleration, $a = g$
• Final velocity = $v$

Using Newton's first equation:

Using Newton's second equation to find distance ($h$):

Using Newton's third equation:

Consider a body projected upwards from the ground with an initial velocity $u$. It rises to a height $h$ and returns to the ground after time $t$.

• Initial velocity = $u$
• Acceleration = $-g$ (acts downward)
• At maximum height, velocity = $0$

Using Newton's second equation:

Using Newton's first equation:

Using Newton's third equation:

• Free fall: $v = gt$, $h = \tfrac{1}{2}gt^2$, $v = \sqrt{2gh}$
• Upward motion: $v = u - gt$, $h = ut - \tfrac{1}{2}gt^2$, $v = \sqrt{u^2 - 2gh}$