Lisset throws a softball from a height of 4 meters, with an initial velocity of 20 meters per second at an angle of 45° with respect to the horizontal. When will the ball be at a horizontal distance of 30 meters from Lisset? Round to the nearest tenth.
To find the time when the ball will be at a horizontal distance of 30 meters from Lisset, we need to use the equations of motion for projectile motion.
The given information includes the initial vertical position (height) of the ball, the initial velocity, and the launch angle.
Let's break down the problem into its horizontal and vertical components:
Horizontal Component:
The horizontal motion of the softball is constant because there is no horizontal acceleration. Therefore, the time taken to travel a horizontal distance of 30 meters will be the same as the time taken to reach a maximum height and then fall back down to a height of 4 meters.
Vertical Component:
The vertical motion can be analyzed separately for the upward and downward halves of the trajectory. The time taken to reach the maximum height and the time taken to fall back to a height of 4 meters is the same.
To find the time taken for the horizontal distance of 30 meters, we need to first calculate the time of flight (total time) for the ball to travel its entire trajectory.
To find the time of flight, we can use the formula:
Time = (2 * initial velocity * sin(angle)) / acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2.
Given:
Initial velocity, u = 20 m/s
Launch angle, θ = 45°
Plugging the values into the formula, we get:
Time = (2 * 20 * sin(45°)) / 9.8
Simplifying:
Time = (40 * sqrt(2)) / 9.8
Now, we can use the time of flight to find the horizontal distance traveled by the ball at any given time using the formula:
Horizontal distance = initial velocity * cos(angle) * time
Therefore, the equation for the horizontal distance is:
30 = 20 * cos(45°) * time
Simplifying:
30 = 20 * (sqrt(2)/2) * time
Now we can solve for time:
30 = 10 * sqrt(2) * time
Divide both sides by 10 * sqrt(2):
time = 30 / (10 * sqrt(2))
Using a calculator, we can evaluate this expression to find the time to the nearest tenth:
time ≈ 1.06 seconds
So, the ball will be at a horizontal distance of 30 meters from Lisset after approximately 1.1 seconds.