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Question

How far will a stone travel over level ground if it is thrown upward at an angle of 26.0 degrees with respect to the horizontal and with a speed of 14.0 m/s?

Answer 1

We can use trigonometry to determine the vertical and horizontal components of the stone's velocity. This will let us determine how far it goes.

Imagine a right triangle with a hypotenuse of 14 units and one angle of 26 degrees. The side opposite the 26 degree angle will be as long as the stone's vertical velocity. The side adjacent to the 26 degree angle will be as long as the stone's horizontal velocity.

Since:

sinA = opposite / hypotenuse...
sin26 = vertical velocity / 14...
vertical velocity = 6.14m/s

cosA = adjacent / hypotenuse;
cos26 = horizontal velocity / 14...
horizontal velocity = 12.56m/s

If the object had an initial vertical velocity of 6.14m/s, we know how long it stayed in the air because:
acceleration = change in velocity / time
Which is the same as saying:
time = change in velocity / acceleration
This applies in any situation where the acceleration is constant.
So, if we want to know how long it took for the stone to reach 0m/s vertical velocity (the point at which it stops rising and starts falling), the change in velocity will be 6.14m/s. If we just divide this number by acceleration due to Earth's gravity (9.81m/s^2), we get 0.63 seconds. The stone traveled upward for 0.63 seconds. This means that it must have also traveled downward for 0.63 seconds before hitting the ground, for a total flight time of 1.26 seconds.

Now, we have a simple velocity equation. Since we know that:
distance = velocity * time
Since we're only worried about how far the stone traveled horizontally, we use the stone's horizontal velocity: 12.56m/s. Time will be the total time that the stone was moving: 1.26sec.

So...
distance = 12.56 * 1.26 = 15.83 meters.

Hope that helped!

Answer 2

Given: initial speed=14m/s at an angle of 26 degrees with respect to the horizontal.

First determine the time it takes the stone to reach the top of its flight using the formula:

v=u+at

where v is the vertical component of the speed at the top(this is equal to 0), u the initial vertical component of the speed, a is the acceleration of gravity (-9.8m/s^2 because the stone is moving against gravity), t is the time it takes for the stone to reach the top. Substitute known values:

0=14sin26+(-9.8)t
t=14*0.438/9.8
=0.62s

The time it takes the stone to complete its flight is twice the time it takes to reach the top of its flight. Using the formula distance=speed * time, we get

distance=14cos26*0.62*2
=14*0.899*0.62*2
=15.6m

Note that we used the horizontal component of its speed (14cos26) to get the distance.

Answer 3

15.73 meters

Answer 4

is that a homework qustion?