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 find the x value at which the tennis ball hits the ground, we need to determine the time it takes for the ball to reach the ground.
We can start by breaking down the vertical motion of the ball. The initial vertical velocity of the ball is 20 m/s (as it is shot backward by the firetruck) and it experiences an acceleration due to gravity of -9.8 m/s².
Using the equation of motion for vertical motion:
y = y0 + v0y * t + (1/2) * a * t²,
where:
y is the vertical position of the ball,
y0 is the initial vertical position (1.4 m),
v0y is the initial vertical velocity (20 * sin(39°)), and
a is the acceleration due to gravity (-9.8 m/s²),
we can solve for the time it takes for the ball to hit the ground.
Rearranging the equation, we get:
0 = 1.4 + (20 * sin(39°)) * t - (4.9) * t².
This is a quadratic equation in terms of t. We can solve it by setting the equation equal to zero and using the quadratic formula:
t² - (20 * sin(39°)) * t + 1.4 = 0.
Plugging in the values, we get:
t² - (20 * sin(39°)) * t + 1.4 = 0.
Using the quadratic formula:
t = [-(-20 * sin(39°)) ± √((-20 * sin(39°))² - 4 * 1 * 1.4)] / (2 * 1).
Simplifying:
t = [20 * sin(39°) ± √((20 * sin(39°))² - 4 * 1 * 1.4)] / 2.
Calculating the values in the square root:
t ≈ [20 * sin(39°) ± √(400 * sin²(39°) - 5.6)] / 2.
The positive root will give us the time taken for the ball to hit the ground. Let's calculate it:
t ≈ [20 * sin(39°) + √(400 * sin²(39°) - 5.6)] / 2.
Using a calculator, we find:
t ≈ 0.973 s.
Now that we have the time it takes for the ball to hit the ground, we can find the x value. The x component of the ball's velocity is the same as the firetruck's velocity, which is +20 m/s. Therefore, the x value at which the ball hits the ground is:
x = x0 + v0x * t,
where:
x0 is the initial x position (63 m),
v0x is the initial x velocity (20 m/s), and
t is the time taken for the ball to hit the ground (0.973 s).
Plugging in the values, we get:
x = 63 + 20 * 0.973.
Calculating:
x ≈ 63 + 19.46.
Therefore, the x value at which the tennis ball hits the ground is approximately 82.46 m.