Julie throws a ball to her friend Sarah. The ball leaves Julie's hand a distance 1.5 meters above the ground with an initial speed of 10 m/s at an angle 38 degrees; with respect to the horizontal. Sarah catches the ball 1.5 meters above the ground.
and the question is? ...
To solve this problem, we can use the equations of motion in projectile motion to find the time of flight and the horizontal distance traveled by the ball.
Let's break down the given information:
- The initial vertical position of the ball when it leaves Julie's hand is 1.5 meters above the ground.
- The initial speed of the ball is 10 m/s.
- The angle between the initial velocity vector and the horizontal is 38 degrees.
- The final vertical position of the ball when Sarah catches it is also 1.5 meters above the ground.
First, let's find the time of flight. To do this, we need to find the time it takes for the ball to reach its highest point.
Step 1: Find the initial vertical velocity component:
The initial vertical velocity component (Viy) can be found using the equation:
Viy = Vi * sin(θ)
where Vi is the initial speed (10 m/s) and θ is the angle (38 degrees).
Viy = 10 m/s * sin(38 degrees)
Viy ≈ 6.08 m/s (rounded to two decimal places)
Step 2: Find the time to reach the highest point:
The time to reach the highest point (t_peak) can be found using the equation:
t_peak = Viy / g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
t_peak = 6.08 m/s / 9.8 m/s²
t_peak ≈ 0.62 seconds (rounded to two decimal places)
Step 3: Find the total time of flight:
The total time of flight (t_total) can be found using the equation:
t_total = 2 * t_peak
t_total = 2 * 0.62 seconds
t_total ≈ 1.24 seconds (rounded to two decimal places)
Now let's find the horizontal distance traveled by the ball.
Step 4: Find the initial horizontal velocity component:
The initial horizontal velocity component (Vix) can be found using the equation:
Vix = Vi * cos(θ)
where Vi is the initial speed (10 m/s) and θ is the angle (38 degrees).
Vix = 10 m/s * cos(38 degrees)
Vix ≈ 7.94 m/s (rounded to two decimal places)
Step 5: Find the horizontal distance:
The horizontal distance (d) can be found using the equation:
d = Vix * t_total
d = 7.94 m/s * 1.24 seconds
d ≈ 9.86 meters (rounded to two decimal places)
Therefore, the ball travels approximately 9.86 meters horizontally before Sarah catches it.