How do you answer this projectile motion problem using vectors?
A 48 feet tall tree stands directly between the pin and the golf ball. The pin is 225 feet away from the golf ball and the tree is 100 feet from the golf ball. A man hits the ball toward the pin with a speed of 80 feet/s at an angle of π/4 rad. Show that the ball avoids hitting the tree and find out if the man gets a hole in one.
One Response
getfreducated
10 Feb 2010
The velocity vector (at t=0) is given by V0cosθ i + V0sinθ j (where i and j are unit vectors parallel to the x and y coordinate axes). The vector changes in time…
V(t) = Vcosθ i + (Vsinθ -gt) j The position vecor is the integral of the velocity vector, so
s(t) = Vcosθ t + (Vsinθ t -1/2 gt^2) j Plugging in our data…
s(t) = 40 √2 t i + (40 √2 t – 16 t^2) j
When the x value of this vector is 100, t=1.77 s which gives a y position of 150 feet > 48 thus the ball clears the tree.
When the y position is 0, t = 3.53 sec and the x position at that time is 200 feet (a respectable shot but not a hole out).
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Reality check…
Range = V^2 cos (90) /g = 200 m. so the second part checks using the "non-vector" approach.
-Fred
That is the method, please check the algebra and aritmatic before submitting.