A ball is thrown off a 30 m high cliff at 2 m/s, 50 degrees above the horizontal. How long does it take to hit the ground?
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To find how long it takes for the ball to hit the ground, we can break down the motion into horizontal and vertical components and determine the time separately for each component.
First, let's analyze the horizontal motion. The initial velocity in the horizontal direction is given as 2 m/s, and the angle above the horizontal is given as 50 degrees. We can use trigonometry to find the horizontal component of the velocity.
Horizontal velocity (Vx) = initial velocity (2 m/s) * cos(angle)
Horizontal velocity (Vx) = 2 m/s * cos(50 degrees)
Next, let's analyze the vertical motion. The initial velocity in the vertical direction is also given as 2 m/s, and the angle above the horizontal is 50 degrees. We can use trigonometry to find the vertical component of the velocity.
Vertical velocity (Vy) = initial velocity (2 m/s) * sin(angle)
Vertical velocity (Vy) = 2 m/s * sin(50 degrees)
Now, we can find the time it takes for the ball to reach the ground by using the vertical motion. We know that the ball is initially at a height of 30 m, and we need to find the time it takes for the ball to reach a height of 0 m (i.e., the ground).
The equation we can use is:
Vertical displacement (Δy) = initial vertical velocity (Vy) * time (t) + 0.5 * acceleration due to gravity (g) * time (t)^2
Since the ball is thrown vertically downward, the initial vertical velocity (Vy) is negative (-2 m/s) and the gravitational acceleration (g) is approximately -9.8 m/s² (taking into account its downward direction). The equation becomes:
0 m = -2 m/s * t + 0.5 * (-9.8 m/s²) * t^2
Now we can solve this quadratic equation for t. Rearranging the terms, we get:
-4.9t^2 - 2t = 0
Factoring out t, we have:
t * (-4.9t - 2) = 0
Setting each factor equal to zero, we find two possible solutions:
t = 0 (which corresponds to the starting time of the motion) or
-4.9t - 2 = 0
Solving the equation -4.9t - 2 = 0, we get:
-4.9t = 2
t = 2 / -4.9 ≈ -0.408
Since time cannot be negative in this context, we disregard the negative solution. Thus, the time it takes for the ball to hit the ground is approximately 0.408 seconds.