(a)No one interferes with the ball. How long does it take the ball to hit the ground? (b)How far did the ball travel horizontally?

Question

Aurora hit a baseball with an initial velocity of 70 feet per second at an angle of 30° with the horizontal. The ball hit her bat when the ball was 3 feet above the ground.

Use answer from part a in calculations for part b

Answer 1

Considering ONLY the y-direction (up/down), the velocity in the y direction (v_y) is calculated using sin(30°) = opposite/hypotenuse = v_y/70 therefore v_y = 70sin(30°) = 35 ft/s.

The equation that relates initial velocity, final velocity, time, and acceleration is v_yf = v_yi + at where "a" in this case is the acceleration of gravity (or "g") which is -32.17 ft/s², v_yi is again the initial velocity in the y-direction, "t" is time, and v_yf is the final velocity in the y direction which will be zero since the ball goes up until it instantaneously stops to reverse direction and come back down. So, considering half of the ball's trip (from the ground up to the peak of it's travel in the y-direction), we can use the equation to solve for time:

v_yf = v_yi + at

v_yf - v_yi = at

(v_yf - v_yi)/a = t

Plugging in the numbers we get:

t = (0 - 35)/(-32.17)

t = 1.088 seconds

However, the ball still needs to fell back to Earth and will take the exact same time to fall down as it did to rise up. So the total time is 1.088(2) = 2.176 seconds

Now, considering ONLY the x-direction, the velocity in the x-direction can be calculated using cos(30°) = adjacent/hypotenuse = v_x/70 therefore v_x = 70cos(30°) = 60.6218 ft/s. It travels at 60.6218 ft/s for 2.176 seconds (as calculated above) so the distance it travels is (60.6218)(2.176) = 131.9 ft