2.8 Vertical motion under gravity

Vertical motion under gravity is a specific case of one dimensional motion with constant acceleration. Here, acceleration is always directed in vertically downward direction and its magnitude is "g".

As the force due to gravity may be opposite to the direction of motion, there exists the possibility that the body under force of gravity reverses its direction. It is, therefore, important to understand that the quantities involved in the equations of motion may evaluate to positive or negative values with the exception of time (t). We must appropriately assign sign to various inputs that goes into the equation and correctly interpret the result with reference to the assumed positive direction. Further, some of them evaluate to two values one for one direction and another of reversed direction.

As pointed out earlier in the course, we must also realize that a change in reference direction may actually change the sign of the attributes, but their physical interpretation remains same. What it means that an attribute such as velocity, for example, can be either 5 m/s or -5 m/s, conveying the same velocity. The interpretation must be done with respect to the assigned positive reference direction.

Velocity

Let us analyze the equation "v = u + at" for the vertical motion under gravity with the help of an example. We consider a ball thrown upwards from ground with an initial speed of 30 m/s. In the frame of reference with upward direction as positive,

$$\begin{array}{l}u=30m/s\mathrm{and}a=-g=-10m/s^2\end{array}$$

Putting this value in the equation, we have :

v = 30 – 10 t

The important aspect of this equation is that velocity evaluates to both positive and negative values; positive for upward motion and negative for downward motion. The final velocity (v) is positive for t<3 seconds, zero for t = 3 seconds and negative for t>3 seconds. The total time taken for the complete up and down journey is 3 (for upward motion) + 3 (for downward motion) = 6 seconds.

The velocities of the ball at successive seconds are :

---------------------------------- Time (t) Final velocity (v)in seconds in m/s ----------------------------------0.0 30 1.0 202.0 10 3.0 04.0 -10 5.0 -206.0 -30 ----------------------------------

The corresponding velocity – time plot looks like as shown in the figure.

We notice following important characteristics of the motion :

1: The velocity at the maximum height is zero (v=0).

2: The time taken by the ball to reach maximum height is obtained as :

$$\begin{array}{l}\mathrm{For}v=0,\\ v=u+at=u-gt=0\\ \Rightarrow u=gt\\ \Rightarrow t=\frac{u}{g}\end{array}$$

3: The ball completely regains its speed when it returns to ground, but the motion is directed in the opposite direction i.e.

$$\begin{array}{l}v=-u\end{array}$$

4: The time taken for the complete round trip is :