A baseball is hit at a height of 1.1 m with an unknown initial velocity at 38 ° above the horizontal. It just clears a barrier of height 36.1 m at a horizontal distance of 63.0 m. Find:
a) the initial speed;
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To find the initial speed of the baseball, we can use the principle of projectile motion and apply the equations of motion for both the horizontal and vertical components. Let's break down the problem step by step.
Step 1: Break the initial velocity into its horizontal and vertical components.
The initial velocity of the baseball can be broken down into two components: the horizontal component (Vx) and the vertical component (Vy). Given that the angle of projection is 38° above the horizontal, we can calculate these components using trigonometry:
Vx = initial speed * cos(angle)
Vy = initial speed * sin(angle)
Step 2: Calculate the time taken for the baseball to reach the barrier height.
To find the time taken for the baseball to reach the barrier, we can use the equation of motion for vertical displacement:
h = Vy * t + (0.5) * g * t^2
Where:
h = vertical displacement (36.1 m)
Vy = vertical component of initial velocity (from Step 1)
t = time of flight (unknown)
g = acceleration due to gravity (approximately 9.8 m/s^2)
Step 3: Calculate the horizontal distance covered during the time of flight.
Since horizontal speed remains constant throughout the motion, we can calculate the horizontal distance covered using the equation:
d = Vx * t
Where:
d = horizontal distance (63.0 m)
Vx = horizontal component of initial velocity (from Step 1)
t = time of flight (from Step 2)
Step 4: Solve for the initial speed.
Now that we have the values for the vertical and horizontal components of the initial velocity, as well as the time of flight and the horizontal distance covered, we can use the equation:
d = Vx * t
to solve for the initial speed.
Plug in the values obtained in Step 3:
63.0 m = Vx * t
Solve for t:
t = 63.0 m / Vx
Substitute the value of t in the equation from Step 2:
36.1 m = Vy * (63.0 m / Vx) + (0.5) * g * (63.0 m / Vx)^2
Now, substitute the values of Vx and Vy from Step 1:
36.1 m = initial speed * sin(angle) * (63.0 m / (initial speed * cos(angle))) + (0.5) * g * (63.0 m / (initial speed * cos(angle)))^2
Simplify and solve for initial speed.