A firetruck is traveling at a velocity of +20 m/s in the x direction. As the truck passes x=63m it shoots a tennis ball backwards with a speed of 20 m/s relative to the truck at an angle of 39° from a height of 1.4 m. Neglect air resistance. At what x value does the tennis ball hit the ground?
To determine the x value at which the tennis ball hits the ground, we need to find the time it takes for the ball to reach the ground. We can do this by analyzing the vertical motion of the ball and using the kinematic equation for vertical motion.
Let's break down the given information:
Initial vertical velocity (Vy0) = 20 m/s * sin(39°)
Vertical displacement (Δy) = -1.4 m (negative because the height is decreasing)
Acceleration due to gravity (g) = -9.8 m/s^2 (negative because it acts downward)
Using the kinematic equation for vertical motion:
Δy = Vy0 * t + 0.5 * g * t^2
Substituting the given values:
-1.4 = (20 * sin(39°)) * t + 0.5 * (-9.8) * t^2
Simplifying the equation:
-1.4 = 12.54 * t - 4.9 * t^2
Rearranging the equation to the quadratic form:
4.9 * t^2 - 12.54 * t - 1.4 = 0
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
a = 4.9, b = -12.54, c = -1.4
Plugging in the values, we get:
t ≈ 0.439 s (ignoring the negative solution)
Now, we can find the x value at which the ball hits the ground. Since the horizontal motion of the ball is independent of the vertical motion, the x velocity remains constant at 20 m/s. We can use the formula:
x = Vx * t
Substituting the values:
x = 20 m/s * 0.439 s
x ≈ 8.78 m
Therefore, the tennis ball hits the ground at an x value of approximately 8.78 meters.