How High is the Arrow at 150 Meters Downrange?

[physicnow]

Hi,
I have an problem that I can not get the answer, any help would be greatly appreciated.

An archer shoots an arrow with a velocity of 45m/s at an angle of 50 degrees with the horizontal. What is the height of the arrow at a point 150 meters downrange?

 

[Astronuc]

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[m2003]

Hello, thank you for reaching out. To solve this problem, we can use the equations of projectile motion. We know that the horizontal and vertical components of velocity are independent of each other, so we can break down the initial velocity of 45m/s into its horizontal and vertical components. The horizontal component will remain constant at 45m/s, while the vertical component will change due to the acceleration of gravity.

Using the equation h = h0 + v0t + 1/2at^2, where h is the height, h0 is the initial height (which we can assume to be 0), v0 is the initial velocity, t is time, and a is acceleration, we can solve for the time it takes for the arrow to travel 150 meters downrange. Since the horizontal velocity remains constant, we can use the equation d = v0t, where d is distance and v0 is the horizontal velocity, to solve for time.

t = d/v0 = 150m/45m/s = 3.33s

Now, we can use this time in the equation for vertical displacement to solve for the height of the arrow at this point:

h = v0t + 1/2at^2 = (45m/s)(3.33s) + 1/2(-9.8m/s^2)(3.33s)^2 = 149.85m

Therefore, the height of the arrow at a point 150 meters downrange is 149.85 meters. I hope this helps!