00:01
A batter hits a ball with an initial horizontal speed of 73 miles an hour and an initial climb rate of 62 miles an hour.
00:06
First we want to draw the vectors involved showing the baseball speed and direction when it leaves the bat.
00:12
So we're going to have a horizontal speed of 73 miles per hour.
00:20
We'll draw that vector.
00:22
And then we have the climb rates, the vertical speed of 62 miles per hour.
00:28
We'll draw that vector right here at the head of this vector.
00:32
Going up which is vertical and that one is 62 miles per hour and then the actual direction that the ball travel would be if we start at the tail of this vector and go to the head of the second vector the ball's vector that it will travel in and we have this angle we're gonna want to find and we're gonna want to find the magnitude of this vector since part b says how fast the ball moving when it leaves the bat so it that would just be the length of the hypotenuse of this right triangle so that's the magnitude of this vector we just call it v so v squared would be 73 square plus 62 square then b would be the square root of 73 square plus 62 square and we can run that through a calculator the square root of 73 squared plus 62 squared and that gives gives us about 95.
01:37
If we rounded the nearest total number, about 96 miles per hour.
01:42
So about 96 miles per hour.
01:48
And then when the ball leaves the bat, find the angle it initially formed with the horizon.
01:52
So make sure we're in degree mode in our calculator.
01:57
And we want to find what is that angle? we can use an inverse tangent.
02:04
That angle would be the inverse tangent.
02:06
Of the opposite side of that angle which is 62 over the adjacent side to 73 and we'll flow that in the calculator and the inverse tangent of 62 over 73 is about a 40 degree angle about 40 degrees if the ball lands 2 .896 seconds after being hit how many feet did the ball travel horizontally so the horizontal component of the velocity 73 miles per hour would not the 62 miles per hour that's going to decrease due to gravity, but the ball is going to travel at 73 miles an hour horizontally for that entire time.
02:51
So we know there are 5 ,280 feet in one mile and 3 ,600 seconds in one hour.
03:07
So first let's just find the distance travel the distance would be the rates or which was given to be 73 miles per hour.
03:21
And we multiply that by the time, which is 2 .896 seconds.
03:27
So first we're gonna convert this to miles or feet per second.
03:32
So if 73 over one, it's miles per hour.
03:38
Let's multiply that by one hour over 3 ,600 seconds.
03:47
Times times 5 ,280 feet over one mile.
03:58
That'll cancel out the miles and that'll cancel out the hours.
04:04
They also need to multiply that 2 .896 seconds.
04:09
So we have 73 times 5 ,280 times 2 .896.
04:18
All that would be over 3 ,600 and here the second units would cancel also and we're just left with feet so let's plug it in the calculator we have 73 times 5 ,280 times 2 .896 and we divide all that by 3 ,600 it gives about 310 feet about 310 feet is how far the ball travels horizontally.
04:54
The next problem says a plane flies with a constant speed of 500 miles per hour in the direction 10 degrees north of west toward its destination.
05:02
The plane encounters a wind blowing at 35 miles per hour in the direction 75 degrees north of east.
05:08
So first we're going to draw the vector diagram for this situation.
05:12
So we'll get a coordinate plane with this being north, here's east, south, and here's west.
05:20
So, so the planes flying in the direction of 10 degrees north of west.
05:27
So that would be in this direction.
05:34
And it's going at 500 miles per hour.
05:38
And that angle again, this angle is 10 degrees north of west.
05:44
And then the plane encounters a wind blowing at 35 miles per hour in the direction of 75 degrees north of east...