a cricketer throw the ball to have maximum horizontal range of 120m.if he throws the ball vertically with same velocity what is the maximum height it can reach?
To find the maximum height a ball can reach when thrown vertically with the same velocity as in the horizontal throw, we can use the principles of projectile motion.
In projectile motion, the vertical and horizontal components are independent of each other. So, the time of flight and initial velocity in the vertical direction will be the same as in the horizontal direction.
The maximum height can be determined using the formula for vertical displacement:
H = (V^2 * sin^2θ) / (2g)
where,
H = maximum height
V = initial velocity in the vertical direction
θ = launch angle (in this case, 90 degrees for a vertical throw)
g = acceleration due to gravity (around 9.8 m/s^2)
Since the initial velocity in the vertical direction is the same as the initial velocity in the horizontal direction, we can substitute the value of V with the vertical component of the initial velocity.
Now, let's calculate the maximum height:
1. First, we need to determine the initial velocity in the horizontal direction.
To obtain the maximum horizontal range of 120m, we can use the formula for horizontal displacement:
R = V^2 * sin(2θ) / g
where,
R = horizontal range
V = initial velocity in the horizontal direction
θ = launch angle
Since θ = 45 degrees for maximum horizontal range, we can rearrange the formula to solve for V:
V = sqrt((R * g) / sin(2θ))
Plugging in the values, we get:
V = sqrt((120 * 9.8) / sin(90)) ≈ 15.52 m/s
2. Next, we can find the maximum height using the formula mentioned earlier:
H = (V^2 * sin^2θ) / (2g)
Since θ = 90 degrees for a vertical throw, we have:
H = (15.52^2 * sin^2(90)) / (2 * 9.8)
H = (240.5504 * 1) / 19.6
H ≈ 12.28 m
Therefore, the maximum height the ball can reach when thrown vertically with the same velocity is approximately 12.28 meters.
Maximum horizontal range is achieved when thrown at an angle of 45 degrees, and equals Vo^2/g. (Vo is the initial velocity). Maximum height when thrown verticslly is
Vo^2/(2g), half as much as maximum range.