Problem 85.

A baseball player hits a homerun, and the ball lands in the left field seats, which is 120 m away from the point at which the ball was hit. The ball lands with a velocity of 20 m/s at an angle of 30 degrees below horizontal. Ignoring air resistance

(A) find the initial velocity and the angle above horizontal with which the ball leaves the bat;

(B) find the height of the ball relatively to the ground.

Solution:

(A) Initial velocity.

Without air resistance this is simple projectile motion. In the present problem we do not know initial velocity: we do not know the magnitude of the velocity (speed) and we do not know its direction.

There are two sets of equations, which describe the motion of the projectile (ball).

Set 1: motion along horizontal axis (axis x – see figure). This is the motion with constant velocity. There is only one equation, which describe this motion:

................................................(1)

Here .

Since the motion along the axis x is the motion with constant velocity then the x-component of the velocity is constant. We know the velocity at the final point. Then we can find the x-component of the velocity at the final point:

This x-component of the velocity is equal to the x-component of the initial velocity:

...........(2)

We also know the x-coordinate of the final point (point B): it is 120 m. We substitute this value in equation (1) and obtain

From this equation we can find the time of travel from point A to point B:

Now we need to analyze the second set of equations.

Set 2: motion along vertical axis (axis y – see figure). This is the motion with constant acceleration – free fall motion. There are three equations, which describe this motion. Only two equations are independent, but it is convenient to write all three equations:

.............(3)

................................................(4)

Since the initial y-coordinate is zero, then

.............................................(5)

We know the y component of the final velocity

This is the y-component of the velocity at the moment of time . We substitute these values in equation (4) and obtain

From this equation we can find the y-component of the initial velocity:

Finally we know the x- and y-components of the initial velocity:

From these expressions we can find the magnitude of the initial velocity and the direction (angle) of the initial velocity:

Now we know the initial velocity.

(B) Final height.

We need to find the final height of the ball (the final y-coordinate). To find the final height we can use equation (3). We just need to substitute the y-component of the initial velocity and the traveled time in this equation: