Physics Problem

Question

Base ball thrown at theta = 37 degrees at 20 m/s, (a) Calculate max height (b) time to reach max height and max horizontal distance (range).

Answer 1

First, find the horizontal and vertical components of the initial velocity. Assuming that angle is measure from the horizontal, then the vertical component is 20*sin(37), and the horizontal is 20*cos(37.

Next, find the time to the top of the trajectory, where v = 0. The change in velocity is a*t, where a is the acceleration of gravity: 9.81m/sec^2

the distance to the top of the trajectory is the average velocity times the time.

Answer 2

Solving this problem comes from breaking down our motion into two separate dimensions, horizontal(x) and vertical(y).

Each of these components can be found by using right angle trigonometry.

So let there be a right triangle with a hypotenuse with a value of 20 and an angle of 37 degrees.

For the h_max and t_max we need to evaluate the vertical direction. So with our triangle we need to use SIN(x) = opposite/hypotenuse.

SIN(37) = y/20

20*SIN(37) = y

y = 12.03630 m/s

Now we can set that value, lets round to 12 m/s, to the initial velocity in the y direction v_yi

Remembering that in the y direction our acceleration due to gravity is -9.8 m/s² and that when an object reaches its maximum height its velocity (v_yf) is 0 m/s

So lets plug into the following formula for motion: v_f = v_i + at

0 m/s = 12 m/s + (-9.8 m/s²)t

-12 m/s = (-9.8 m/s²)t

t = 1.2 seconds

This value is the time to reach the maximum height.

We can then use that to find the maximum height with the following equation: y = v_it + (1/2)at²

Let's plug in:

y = (20 m/s)*(1.2 s) + 0.5*(-9.8m/s²)*(1.2 s)²

y = 16.9 m

Now for the range we need to use the following trig function on the original right triangle we set up:

COS(x) = adjacent/hypotenuse

COS(37) = x/20

20*COS(37) = x

x = 16 m/s

Now since there is no acceleration in the x direction we can use the following: v = d/t

LEt's plug in:

20 m/s = x/1.2s

x = (20 m/s)*(1.2 s)

x = 24 meters