A basketball player wants to score a three point shot from the 3 point line. The

Question

A basketball player wants to score a three point shot from the 3 point line. The ball has a circumference of 749 mm, and a mass of 624g. (NBA spec) The three point line is 6.7 m from the goal. The goal is 3.05 m high and has a diameter of .45m. The player is 2 m tall and has 1m long arms and weighs 150N. If he shoots from a height equal to the height of the rim at the edge of the 3 point line:

c) Over his 1 meter of arm length what work and energy is placed into the ball?

•d) If there are four revolutions per second of back spin on the ball how much rotational energy is there?

•e) How much mechanical energy is there in the ball as the ball is released; (Total Kinetic) linear plus rotational?

•f) What is the angle of opportunity or to what precision must the shot be made plus or minus on the horizontal plane, or from right to left not up and down?

g) What is the angular acceleration is needed to induce the 4 rev/sec rotation of the ball, and what centripetal acceleration is developed?

Explanation / Answer

a)There is no acceleration in the x direction, and g is acting in the y direction. Choose the point of the shot to be (0,0) and the direction of the goal to be the positive x direction.

X=X0+V0cost; Y=Y0+V0sint-gt2/2

y=0. x=6.7m

6.7=V0cost; 0=V0sint-gt2/2

Now there are 2 equations and 2 unknowns, you should be able to solve for both. There will be a quadratic formula involved. The best way to proceed is to eliminate t and solve for the value of a trig function after using identities to get an equation with only one type of trig function (as in 0=atan2+btan+c solve for tan) Then use inverse trig to get . Finding V0 is somewhat easy after that.

b) Now that you have V0, use V0=at+Vi. (Vi=0 btw)

c)Now that you have a (acceleration), use F=ma to get force. Then use W=F*d to get the work done on the ball.

d)4rev/s(2/rev)=8 rad/s= (angular velocity)

ER=(1/2)I2 where I is the moment of inertia for a hollow sphere (A sphere has I=2/5mr2, you will need to look up what I is for a hollow sphere.

e) Total E=K+ER=(1/2)mV02+(1/2)I2

f) Angle of opportunity. The rim has a certain diameter. The ball has a smaller diameter but must fit inside the rim. The allowable horizontal distance (call it p) will be the difference between these. The angle will be =tan-1(p/x)

g)Since you have a (accel) of the ball to get V0, you can calculate the time it takes to accelerate it over 1m. Use V=at. This t is the same t that is required to induce the angular velocity.

=I where I is the moment of inertia and is the angular acceleration.

For circular motion, centripetal acceleration is a=V2/r=(r)2/r=r2

I hope this helps.

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