Question

A baseball player is 98m away from home plate when he throws the ball at a 30° angle above horizontal from a height of 1.45 m (the height of his hand).

a. What is the initial speed of the throw for the ball to land exactly on home plate? In order to solve this problem, set up equations for x and y position vs time. Put in all known values. You will have two remaining unknowns: v0(the magnitude of the initial velocity) and t. Solve the system of equations.

b.How long does it take the ball to get to home plate?

c.What is the maximum height that the ball is above the ground?

d. Sketch with coordinate system

Answer 1

(a), (b) The x-component of the initial velocity is

The y-component of the initial velocity is

The ball is under gravitational acceleration of the Earth, the acceleration in the x-direction is zero and the acceleration of the ball in the y-direction is

The x-component of velocity does not change with time

The horizontal distance covered by the ball is given by

If the ball reaches the home plate in time T, the horizontal distance covered by the ball in this time is

Given the horizontal distance covered by the ball is 98 m we get

The acceleration in the y-direction is constant. The y-motion is goverened by the kinematics equations for uniform motion. We use

The y-coordinate when the ball reaches the home plate is zero.

Using equation (1)

Solving equation (1) for initial speed

(c) At the maximum height the ball changes the direction of its vertical velocity from up to down, the vertical velocity of the ball is momentarily zero at this point. Using the kinematics equation

We get

(d) The sketch of the ball trajectory with the coordinate system is