Chapter 2: Q8. (page 42)

A ball is thrown downward at a speed of $20 m / s$ . Choosing the +y axis pointing up and neglecting air resistance, which equation(s) could be used to solve for other variables? The acceleration due to gravity is localid="1642661589470" $9.81 m / s^{2}$ downward.
(a) $v = (20 m / s) - g t$ .
(b) $y = y_{0} + (- 20 m / s) t - \frac{1}{2} g t^{2}$ .
(c) $v^{2} = {(20 m / s)}^{2} - 2 g (y - y_{0})$ .
(d) $(20 m / s) = (v + v_{0}) / 2$ .
(e) All of the above.

Short Answer

Expert verified

The correct option is (b) $y = y_{0} + (- 20 m / s) t - \frac{1}{2} g t^{2}$ and (c) $v^{2} = (20 m / s) - 2 g (y - y_{0})$

Step by step solution

Step 1. Definitions of displacement and velocity of the ball

Displacement gives the shortest length of any path between two points. The two points are the initial as well as final points of any path covered by the ball.

Velocity gives the ratio of the displacement and the time covered by the ball to reach its endpoint.

Step 2. Analysis of option (a) v=20 m/s-gt

Initial velocity of the ball moving downward, $u = - 20 m / s$

Acceleration due to gravity in the downward directionlocalid="1642661856728" $= 9.81 m / s^{2}$ .

Choose the +y-axis direction and neglect the air resistance.

The equation thus formed is:

localid="1642663931454" $v = (20 m / s) - g t$

Here, the initial velocity of the ball is $u$ with the value $20 m / s$ . It is positive as the ball is moving in the opposite direction. But it is given that the ball is moving downward with velocity localid="1642663920339" $u (- 20 m / s)$ .

So, option (a) is incorrect.

Step 3. Analysis of option (b) y=y0+-20 m/st-12gt2

The required equation can be written as:

$y = y_{0} + (- 20 m / s) t - \frac{1}{2} g t^{2}$

Solve the above equation further.

$\begin{matrix} y = y_{0} + (- 20 m / s) t + \frac{1}{2} (- g) t^{2} \\ = y_{0} + (- 20 m / s) t + \frac{1}{2} (- 9.81) t^{2} \end{matrix}$

Here, the initial distance covered by the ball is $y_{0}$ , the final distance is $y$ , the initial velocity of the ball acting downward is $u (- 20 m / s)$ , and the acceleration due to gravity acting downward is $(- g) (- 9.81 m / s^{2})$

This equation satisfies the requirement and can be used to solve other variables.

Thus, option (b) is correct.

Analysis of option (c) v2=20 m/s2-2gy-y0

The required equation is:

$v^{2} = {(20 m / s)}^{2} - 2 g (y - y_{0})$

This equation can be further solved as:

$\begin{matrix} v^{2} = {(- 20 m / s)}^{2} + 2 (- g) (y - y_{0}) \\ = {(- 20 m / s)}^{2} + 2 (- 9.81 m / s^{2}) (y - y_{0}) \end{matrix}$

Here, the final distance covered by the ball is $y$ , the initial distance traveled by the given ball is $y_{0}$ , the initial velocity of the same ball acting downward is $u (- 20 m / s)$ , and the acceleration due to gravity acting downward is $(- g) (- 9.81 m / s^{2})$ .

So, option (c) is correct according to the given conditions. It can be used to calculate the other variables.

Step 5. Analysis of option (d) 20 m/s=v+v0/2 and option (e) All of the above

The required equation is:

$(20 m / s) = (v + v_{0}) / 2$

Here, 20 m/s is the average of velocities $v$ and $v_{0}$ . But it is given that 20 m/s is the initial velocity of the ball.

Hence, option (d) is incorrect.

As options (a) and (d) are incorrect, option (e) is also incorrect.

Step 6. Determination of the correct option

The equations $y = y_{0} + (- 20 m / s) t - \frac{1}{2} g t^{2}$ and $v^{2} = {(20 m / s)}^{2} - 2 g (y - y_{0})$ are correct and can be used to determine other variables.