A 15.0 kg dart is thrown horizontally into a dartboard. The thrower applies a 80.0 N force over a distance of 88.2 cm in order to propel the dart.

a) Calculate the work done by the dart thrower.
b) Determine the velocity of the dart.
c) If the dart went 4.50 cm into the dartboard , find the force the dartboard exerts on the dart

a) W+Fx =80•0.882=70.56 J

b) W=KE =mv²/2
v=sqrt(2W/m) = sqrt(2•70.56/15)=3.07 m/s
c) KE=W(fr) =F•x
F=W(fr)/x=KE/x=F/x = 70.56/0.045=1568 N

a) To calculate the work done by the dart thrower, we can use the formula for work:

Work = Force applied * Distance

In this case, the force applied by the dart thrower is given as 80.0 N and the distance over which the force is applied is 88.2 cm. However, we need to convert the distance into meters to be consistent with the unit of force in Newtons.

1 meter is equal to 100 centimeters, so 88.2 cm = 88.2/100 = 0.882 meters.

Now, we can calculate the work done:

Work = 80.0 N * 0.882 m = 70.56 joules

Therefore, the work done by the dart thrower is 70.56 joules.

b) To determine the velocity of the dart, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.

The work done by the dart thrower (calculated in part a) is equal to the change in kinetic energy of the dart. The initial kinetic energy of the dart is zero since it starts from rest.

So, we can write:

Work = Change in kinetic energy

The kinetic energy of an object can be calculated using the formula:

Kinetic energy = (1/2) * mass * velocity^2

Rearranging the equation for work and kinetic energy, we get:

(1/2) * mass * velocity^2 = Work

Since we know the mass of the dart is 15.0 kg and the work done is 70.56 joules, we can solve for the velocity:

(1/2) * 15.0 kg * velocity^2 = 70.56 J

Dividing both sides by (1/2) * 15.0 kg, we get:

velocity^2 = (70.56 J) / ((1/2) * 15.0 kg)

velocity^2 = 9.408 m^2/s^2

Taking the square root of both sides, we find:

velocity ≈ 3.06 m/s

Therefore, the velocity of the dart is approximately 3.06 m/s.

c) To find the force the dartboard exerts on the dart, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration:

Force = mass * acceleration

In this case, we need to calculate the acceleration of the dart. Since the dart went 4.50 cm into the dartboard, we can use this information to find the acceleration.

The distance traveled by the dart into the dartboard is a measure of displacement. We can use the equation of motion:

Displacement = Initial velocity * time + (1/2) * acceleration * time^2

Given that the initial velocity of the dart is the same as the final velocity (approximately 3.06 m/s) and the displacement is 4.50 cm = 0.045 m, we can rearrange the equation and solve for the acceleration:

0.045 m = (1/2) * acceleration * time^2

Substituting the initial velocity of the dart as 3.06 m/s into time^2, we have:

0.045 m = (1/2) * acceleration * (3.06 m/s)^2

0.045 m = (1/2) * acceleration * 9.3636 m^2/s^2

Simplifying further:

(1/2) * acceleration = 0.045 m / 9.3636 m^2/s^2

(1/2) * acceleration ≈ 0.004815 s^-2

Now we can solve for the acceleration:

acceleration ≈ 0.004815 s^-2 * 2 ≈ 0.00963 m/s^2

Finally, we can calculate the force the dartboard exerts on the dart using Newton's second law:

Force = mass * acceleration

Force = 15.0 kg * 0.00963 m/s^2 ≈ 0.1445 N

Therefore, the force the dartboard exerts on the dart is approximately 0.1445 N.