A boy throws a ball with a horizontal speed of 2.5 m/s at a height of 0.8 m above the ground. The Coefficient Of Restitution between ball and ground is 80%.
• Calculate the x− and y−components of the ball’s velocity, immediately after its first bounce. Correct numerical expressions (put in m/s) suffice for an answer, without calculating their final returns.
• Calculate the ball’s maximum height h, immediately after its first bounce. Express this answer (put in m) with an accuracy of three digits after the floating point.
To calculate the x- and y-components of the ball's velocity immediately after its first bounce, we can use the principles of conservation of energy and conservation of momentum.
1. Calculate the x-component of the velocity (Vx):
Since there is no force acting horizontally, the horizontal velocity remains constant throughout the flight. Therefore, the x-component of the velocity after the bounce is the same as the initial horizontal velocity, which is 2.5 m/s.
Vx = 2.5 m/s
2. Calculate the y-component of the velocity (Vy):
Using the principle of conservation of energy, we can determine the y-component of the velocity after the bounce. The initial potential energy is given by m * g * h, where m is the mass of the ball, g is the acceleration due to gravity (9.8 m/s^2), and h is the initial height (0.8 m). The initial kinetic energy is zero since the ball is only at the height and has no vertical velocity initially.
Since the coefficient of restitution (COR) is 80%, we know that the ball retains 80% of its energy after each bounce.
Using the formula for conservation of energy:
0.8 * m * g * h = 0.8 * m * v^2 / 2
Simplifying and solving for v:
v^2 = 1.6 * g * h
v = sqrt(1.6 * g * h)
Now we can substitute the known values:
v = sqrt(1.6 * 9.8 * 0.8)
v ≈ 3.051 m/s
The y-velocity component (Vy) after the bounce is the positive square root of this value:
Vy ≈ 3.051 m/s
Therefore, the x- and y-components of the ball's velocity immediately after its first bounce are:
Vx = 2.5 m/s
Vy ≈ 3.051 m/s
To calculate the ball's maximum height h immediately after its first bounce, we can use the conservation of energy.
Using the formula for conservation of energy:
Potential energy at maximum height = 0.8 * m * g * h_max = Kinetic energy at the first bounce = 0.5 * m * (Vx^2 + Vy^2)
Since we have already calculated the values of Vx and Vy, we can plug them into the equation:
0.8 * m * g * h_max = 0.5 * m * (2.5^2 + 3.051^2)
Simplifying:
0.8 * g * h_max ≈ 0.5 * (2.5^2 + 3.051^2)
Now we can solve for h_max:
h_max ≈ 0.5 * (2.5^2 + 3.051^2) / (0.8 * g)
Substituting the known value of g:
h_max ≈ 0.5 * (2.5^2 + 3.051^2) / (0.8 * 9.8)
h_max ≈ 0.5 * (6.25 + 9.308) / 7.84
h_max ≈ 0.5 * 15.558 / 7.84
h_max ≈ 7.779 / 7.84
h_max ≈ 0.991 m
Therefore, the ball's maximum height immediately after its first bounce is approximately 0.991 m.