girl throws the ball with a certain angle. It covers maximum horizontal distance R. How much height ball reaches if she throws the same ball vertically upward? Assume her muscles give the ball same speed in each case.
Maximum range is obtained when throwing at a 45 degree angle, and equals V^2/ g . That is the range, R.
V is the initial velocity of the throw, and g is the acceleration of gravity.
If thrown vertically upward at the same velocity, the height attained is V^2/(2g)' which is R/2.
To determine the height the ball reaches when thrown vertically upward, we can use the concept of projectile motion. When the ball is thrown with a certain angle, it follows a curved path known as a projectile trajectory. The maximum horizontal distance covered by the ball, R, is achieved when the ball is launched at an optimal angle, called the optimal launch angle.
To find the height the ball reaches when thrown vertically upward, we can consider the properties of projectile motion:
1. The initial vertical velocity of the ball when launched vertically upward will be equal to the initial vertical velocity component when it was launched at the optimal launch angle.
2. The time taken for the ball to reach its maximum height when launched vertically upward will be the same as the time taken for it to reach its maximum horizontal distance in the previous case.
3. The height the ball reaches when thrown vertically upward can be determined using equations of motion.
Thus, to find the height:
1. Start by determining the time of flight when the ball is thrown at the optimal launch angle and reaches a maximum horizontal distance, R. You can use the following equation for horizontal motion:
R = (initial velocity * time of flight) * cos(optimal launch angle)
Rearrange the equation to solve for time of flight:
time of flight = R / (initial velocity * cos(optimal launch angle))
2. Now, when the same ball is thrown vertically upward, it will take the same time to reach its maximum height. So, using the time of flight calculated above, we can determine the height it will reach using the following equation for vertical motion:
height = (initial vertical velocity * time of flight) - (0.5 * acceleration due to gravity * (time of flight)^2)
Substitute the known values:
height = (initial vertical velocity * time of flight) - (0.5 * 9.8 * (time of flight)^2)
Since the initial vertical velocity will be the same as the initial vertical velocity component when the ball was launched at the optimal launch angle, you can substitute it with the value obtained in step 1.
3. Solve the equation to find the height the ball reaches when thrown vertically upward.
Remember, the calculations will depend on the specific values you have for the initial velocity and the optimal launch angle.