A footballer kicks a ball at an angle of 45° with the horizontal. If the ball was in air for 10 seconds and lands 4000meters away. Determine the initial speed
Angle of Projection = 45°
Horiontal distance covered by Ball(R) = 4000 m
Time in air = 10 s
R = u² sin 2θ /g where u = Initial Speed, g = Speed of gravity = 10
4000 = u² sin 90 / 10
∴ u² = 40000
∴ u = 200m/s
horizontal distance=V*cos45*time
V=4000/10*cos45=400*1.4=560 m/s a speeding bullet?
4000m is a long way in 10 sec. at 60miles/hr, that is 88ft/sec=about30m/sec
Well, that footballer must have really long legs if they can kick a ball 4000 meters away! Perhaps they should sign up for the long jump instead. But let's put the jokes aside for a second and solve this problem.
To determine the initial speed of the ball, we can break its motion into horizontal and vertical components. The horizontal component will remain constant throughout the motion, while the vertical component will experience free fall due to gravity.
Since the angle of the kick is 45°, the initial velocity in the vertical direction is equal to the initial velocity in the horizontal direction. Let's call this initial velocity "v".
We know that the time of flight is 10 seconds, so the vertical displacement can be determined using the formula:
d = v*t + (1/2)*g*t^2
where d is the vertical displacement, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time of flight.
Since the ball lands at the same height it was kicked from (we assume it's a perfect kick), the vertical displacement d is zero. Plugging in the values, we get:
0 = v*10 + (1/2)*9.8*(10^2)
Simplifying this equation, we get:
490 = 10v
Dividing both sides by 10, we find that:
v = 49 m/s
So, the initial speed of the ball is approximately 49 m/s. That's one fast kick! But remember, in real-life situations, footballers don't usually have superpowers. I hope this answer kicks you in the right direction!
To determine the initial speed of the ball, we can use the horizontal and vertical components of the motion separately.
The horizontal component of the motion remains constant, as there is no horizontal force acting on the ball after it leaves the ground. Therefore, we can assume that the horizontal speed remains the same throughout the motion.
The horizontal distance covered by the ball is given as 4000 meters, and the time of flight is given as 10 seconds. Therefore, we can use the formula:
Horizontal distance = Horizontal speed × time
Substituting in the given values, we have:
4000 meters = Horizontal speed × 10 seconds
Simplifying this equation, we have:
Horizontal speed = 4000 meters / 10 seconds
Horizontal speed = 400 meters per second
Now, let's look at the vertical component of the motion. The initial velocity in the vertical direction is given by the equation:
Vertical initial velocity = Initial speed × sin(angle)
Since the angle given is 45°, we can substitute it into the equation:
Vertical initial velocity = Initial speed × sin(45°)
Since sin(45°) is 1/√2 ≈ 0.707, we can simplify the equation further:
Vertical initial velocity ≈ Initial speed × 0.707
Now, we know that the time of flight is 10 seconds, and the ball reaches its maximum height halfway through the time of flight, so the time to reach maximum height is 5 seconds. At this point, the vertical velocity becomes zero.
Using this information, we can calculate the maximum height using the equation:
Maximum height = Vertical initial velocity × time + (1/2) × acceleration × time^2
Since the vertical initial velocity is zero at the maximum height, the equation simplifies to:
Maximum height = (1/2) × acceleration × time^2
We know that the acceleration due to gravity is approximately 9.8 meters per second squared, so we can substitute this value into the equation:
Maximum height = (1/2) × 9.8 meters per second squared × (5 seconds)^2
Simplifying this equation, we have:
Maximum height = 122.5 meters
Now, we can use the maximum height to calculate the vertical component of the initial velocity:
Vertical initial velocity = 122.5 meters
Finally, substituting this value into the equation:
Vertical initial velocity = Initial speed × 0.707
122.5 meters = Initial speed × 0.707
Simplifying this equation, we have:
Initial speed ≈ 122.5 meters / 0.707 ≈ 173.4 meters per second
Therefore, the initial speed of the ball is approximately 173.4 meters per second.