A ball, with a mass of 0.144 kg, is thrown from the top of a 76.8 m building (point A) at an angle, θ = 65º with respect to the horizontal. The initial speed of the ball is unknown. The ball reaches a maximum height of 12 m above the building after which it hits the ground at point B. The difference between the kinetic energy of the ball at point A and point B, is equal to:
Assume the value of the gravitational acceleration as g = 9,81 ms-2.
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Answer
To find the difference between the kinetic energy of the ball at point A and point B, we need to calculate the kinetic energy at each point and then find the difference.
First, let's find the initial velocity of the ball at point A. We can use the conservation of energy to do this. At point A, the ball is only at a height of 12m above the building, so the potential energy at that point is mgh, where m is the mass of the ball (0.144 kg), g is the acceleration due to gravity (9.81 m/s^2), and h is the height (12 m). Therefore, the potential energy at point A is (0.144 kg)(9.81 m/s^2)(12 m) = 17.1328 J.
The initial kinetic energy can be calculated using the equation KE = (1/2)mv^2, where m is the mass of the ball and v is the initial velocity. Since we don't know the initial velocity, we can rearrange the equation to solve for v: v = sqrt((2*KE)/m). Plugging in the values, we get v = sqrt((2*17.1328 J)/(0.144kg)) = 8.5879 m/s.
Next, let's calculate the velocity of the ball at point B. To do this, we need to find the maximum height reached by the ball above the building. The maximum height can be calculated using the equation h = (v^2*sin^2(θ))/(2*g), where θ is the angle of the throw (65º), and g is the acceleration due to gravity. Plugging in the values, we get h = (8.5879 m/s)^2 * sin^2(65º) / (2 * 9.81 m/s^2) = 2.5986 m.
Now, we can use the equation v^2 = u^2 + 2 * a * s, where u is the initial velocity (8.5879 m/s), a is the acceleration due to gravity (-9.81 m/s^2), and s is the distance traveled (76.8 m + 2.5986 m = 79.3986 m since the ball is thrown 76.8 m horizontally and then reaches a maximum height of 2.5986 m). Solving for v, we get v = sqrt(u^2 + 2 * a * s) = sqrt((8.5879 m/s)^2 + 2 * -9.81 m/s^2 * 79.3986 m) = 24.8989 m/s.
Now, we can calculate the kinetic energy at point B using the equation KE = (1/2)mv^2, where m is the mass of the ball and v is the velocity at point B. Plugging in the values, we get KE = (1/2)(0.144 kg)(24.8989 m/s)^2 = 87.9336 J.
The difference in kinetic energy between point A and point B is then given by 87.9336 J - 17.1328 J = 70.8008 J.
Therefore, the difference between the kinetic energy of the ball at point A and point B is 70.801.