Question

Two students are on a balcony a distance hh above the street. One student throws a ball vertically downward at a speed v0v0; at the same time, the other student throws a ball vertically upward at the same speed.

(a) Write the kinematic equation for the yy -coordinate of each ball.
(b) Set the equations found in part (a) equal to height 0 and solve each for tt symbolically using the quadratic formula. What is the difference in the two balls' time in the air?
(c) Use the time-independent kinematics equation to find the velocity of each ball as it strikes the ground.
(d) How far apart are the balls at a time tt after they are released and before they strike the ground?

Answer 1

The kinematic equation for the vertical coordinate of each ball can be written as: yy = hh + v0t - (1/2)gt^2. The difference in the two balls' time in the air can be found by using the quadratic formula. The velocity of each ball as it strikes the ground can be found using the time-independent kinematics equation.

Step-by-step explanation:

(a) The kinematic equation for the vertical coordinate (yy) of each ball can be written as:

yy = hh + v0t - (1/2)gt^2

Where:

• yy is the vertical coordinate of the ball
• hh is the initial height of the ball above the street
• v0 is the initial velocity of the ball
• t is the time
• g is the acceleration due to gravity

(b) Setting the equations equal to height 0 and solving for t symbolically using the quadratic formula:

0 = hh + v0t - (1/2)gt^2

Using the quadratic formula:

t = (-v0 ± √(v0^2 + 2ghh))/g

The difference in the two balls' time in the air can be found by subtracting the time for the ball thrown upward from the time for the ball thrown downward.

(c) Using the time-independent kinematics equation:

v = v0 - gt

We can find the velocity of each ball as it strikes the ground by substituting t = 0 into the equation.

(d) The distance between the balls at time t after they are released and before they strike the ground can be found by subtracting the horizontal distances traveled by each ball.