Question

A baseball is hit nearly straight up into the air with a speed of 22 m/s.
(a) How high does it go?
m
(b) How long is it in the air?
s

Answer 1

Concepts and reason

The concepts used to solve this problem are gravitational force and acceleration due to gravity and projectile motion.

Initially, how high the baseball goes can be calculated by using the kinematic equation involving initial, final velocities, maximum height reached by the baseball and acceleration due to gravity.

Later, use the information of the maximum height the baseball reached to calculate the duration the baseball was in the air in its upward journey using the kinematic equation involving initial, final velocity, acceleration due to gravity and time.

Finally, total duration of the baseball in air can be calculated by using the idea that the total duration comprised of the time taken to move up and move down by the baseball.

Fundamentals

When an object is thrown in air from the surface of the Earth, it experiences a gravitational force of Earth on it. This gravitational force pulls the object downward. Due to this pull, the object accelerates downward.

Projectile is an object that is thrown in air moves under the action of gravitational force and makes a curved trajectory. Neglecting the air resistance, its path is completely governed by the gravitational force.

Here, the base ball is hit nearly straight up into the air and so it has initial velocity along vertical direction alone. The baseball leaves the ball straight up into the air with initial vertical velocity.

The expression for the maximum vertical distance the baseball covered is,

Here, u is the initial velocity of the baseball, v is the final velocity of the baseball, s is the maximum height of the baseball goes up into the air, and g is the acceleration due to gravity of the earth.

Express the relation to calculate the time taken by the baseball to move maximum height into air in terms of initial velocity, final velocity, and acceleration due to gravity.

$v=u-gt $

Here, t is the time taken by the baseball to move up.

The time duration of the baseball in air to move up and down is twice the time taken by it to either up or down.

Express the relation to find how long the baseball was in air.

Here, T is the total duration of the baseball in air.

(a)

The expression for the maximum vertical distance of the baseball covered is,

Rearrange the above equation in terms of s.

Substitute 22 m/s for u, 0 for v and for g to find s.

(b)

Express the relation to calculate the time taken by the baseball to move maximum height into air.

$v=u-gt $

Rearrange the above equation in terms of t.

Substitute 22 m/s for u, 0 for v, and for g to find t.

The expression to calculate the total duration of the baseball in air is,

Substitute 2.24 s for t to find T.

Ans: Part (a)

The height the baseball goes up into the air is.