A golfer hits a shot to a green that is elevated 2.70 m above the point where the ball is struck. The ball leaves the club at a speed of 19.0 m/s at an angle of 36.0˚ above the horizontal. It rises to its maximum height and then falls down to the green. Ignoring air resistance, find the speed of the ball just before it lands.
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To find the speed of the ball just before it lands, we need to analyze the motion of the ball in two parts: its vertical motion and its horizontal motion. We'll consider the vertical motion first.
1. Vertical Motion:
At the maximum height, the vertical velocity of the ball is momentarily zero. This means that the initial vertical velocity (upwards) is equal to the final vertical velocity (downwards) just before it lands.
Given:
Initial vertical velocity (upwards) = 19.0 m/s * sin(36.0°)
Acceleration due to gravity (g) = 9.8 m/s^2
Height (h) = 2.70 m
Using the kinematic equation:
vf^2 = vi^2 + 2 * a * Δy
Where:
vf = final vertical velocity
vi = initial vertical velocity
a = acceleration due to gravity
Δy = change in height
We can rearrange the equation to solve for vf:
vf = sqrt(vi^2 + 2 * a * Δy)
Substituting the known values:
vf = sqrt((19.0 m/s * sin(36.0°))^2 + 2 * 9.8 m/s^2 * 2.70 m)
Calculating this using trigonometric and arithmetic operations:
vf ≈ 12.81 m/s
So, the final vertical velocity of the ball just before it lands is approximately 12.81 m/s downwards.
2. Horizontal Motion:
We can ignore air resistance, which means there is no horizontal force acting on the ball. Thus, the horizontal velocity remains constant throughout its flight.
Given:
Initial horizontal velocity (vix) = 19.0 m/s * cos(36.0°)
No acceleration horizontally, so the final horizontal velocity (vfx) remains the same as the initial horizontal velocity.
Therefore, the speed of the ball just before it lands is equal to the final horizontal velocity (vfx).
Substituting the known value:
vfx = 19.0 m/s * cos(36.0°)
Calculating this using trigonometric and arithmetic operations:
vfx ≈ 15.21 m/s
Thus, the speed of the ball just before it lands is approximately 15.21 m/s.