You and your friend are practicing pass plays with a football. You throw the football at a 35.0° angle above the horizontal at 19.0 m/s. Your friend starts right next to you, and he moves down the field directly away from you at 5.50 m/s. How long after he starts running should you throw the football? Assume he catches the ball at the same height at which you throw it.
Vo = (19m/s,35 deg.).
Xo = hor.= 19cos35 = 15.56m/s.
Yo = ver.= 19sin35 = 10.90m/s.
h = (Vf^2 - Yo^2) / 2g,
h = (0 - (10.9)^2) / -19.6 = 6.06m.
t(up) = (Vf - Yo) / g,
t(up) = (0 - 10.9) / -9.8 = 1.11s.
t(dn) = t(up) = 1.11s.
T = t(up) + t(dn) = 1.1 + 1.11 = 2.22s
= Time in flight.
d = Xo*T = 15.56 * 2.22 = 34.5m = hor
distance traveled.
t = d/V = 34.5 / 5.5 = 6.27s
td + T = 6.27s,
td = 6.27 - T = 6.27 - 2.22 = 4.07s.
after the receiver starts running, the
ball should be thrown.
To solve this problem, we need to analyze the motion of both the football and your friend. Let's break it down step by step:
1. Calculate the vertical component of the initial velocity of the football.
- The vertical component can be found by multiplying the initial velocity (19.0 m/s) by the sine of the launch angle (35.0°).
vertical component = 19.0 m/s * sin(35.0°)
2. Determine the time it takes for the football to reach the same height where your friend is running.
- Since the height is the same, the vertical distance covered by the football will be equal to zero.
- We can use the formula: vertical displacement = vertical component * time + 0.5 * acceleration * time², where acceleration is due to gravity and equals -9.8 m/s².
0 = vertical component * time + 0.5 * (-9.8 m/s²) * time²
This is a quadratic equation, which we can solve using the quadratic formula: time = (-b ± sqrt(b² - 4ac)) / (2a)
3. Calculate the horizontal distance your friend covers during that time.
- Your friend moves down the field directly away from you at a constant velocity of 5.50 m/s.
- So the horizontal distance covered by your friend will be the product of his velocity and the time calculated in step 2.
horizontal distance = 5.50 m/s * time
4. Find the time it takes for the football to reach your friend.
- The horizontal distance covered by the football should be equal to the horizontal distance covered by your friend.
horizontal distance = 5.50 m/s * time
By following these steps, you can calculate the time after which you should throw the football to your friend.