A person initially at rest throws a ball upward at an angle θ0= 75 ∘ with an initial speed v0=15 m/s . He tries to catch up to the ball by accelerating with a constant acceleration a for a time interval of 1.03 s and then continues to run at a constant speed for the rest of the trip. He catches the ball at exactly the same height he threw it. Let g= 9.81 m/s2 be the gravitational constant. What was the person's acceleration a (in m/s2)?
a=
To find the person's acceleration, we can start by analyzing the motion of the ball when it is in the air. Let's break it down into horizontal and vertical components.
In the vertical direction, the ball moves under the influence of gravity. The initial vertical velocity, vy0, can be calculated using the initial speed v0 and the launch angle θ0. We can use trigonometry to find the initial vertical velocity:
vy0 = v0 * sin(θ0)
The time it takes for the ball to reach its highest point, t_h, can be determined by dividing the initial vertical velocity by the acceleration due to gravity, g:
t_h = vy0 / g
The maximum height reached by the ball, H, can be calculated using the vertical component of the equation of motion:
H = (vy0^2) / (2 * g)
Next, we need to determine how high the person can jump. Considering that the person is standing on the ground, the maximum height the person can reach when jumping vertically is given by:
H_person = (1/2) * a * (t_jump^2)
where a is the acceleration and t_jump is the time the person spends accelerating before continuing to run at a constant speed.
Now, since the person wants to catch the ball at the same height it was thrown, the maximum height reached by the person must be equal to the maximum height reached by the ball:
H_person = H
Substituting the expressions for H_person and H and rearranging the equation, we get:
(1/2) * a * (t_jump^2) = (vy0^2) / (2 * g)
Now, let's plug in the given values and solve for a:
v0 = 15 m/s
θ0 = 75°
t_jump = 1.03 s
g = 9.81 m/s^2
First, calculate vy0:
vy0 = v0 * sin(θ0)
= 15 * sin(75°)
≈ 14.64 m/s
Next, substitute the values:
(1/2) * a * (1.03^2) = (14.64^2) / (2 * 9.81)
Multiply both sides by 2:
a * (1.03^2) = (14.64^2) / 9.81
Divide both sides by (1.03^2):
a ≈ [(14.64^2) / 9.81] / (1.03^2)
a ≈ 29.25 / 1.0609
a ≈ 27.54 m/s^2
Therefore, the person's acceleration, a, is approximately 27.54 m/s^2.