A baseball is hit with an initial speed of 42 m/s at an angle of 30° relative to the x (horizontal) axis.(a) Find the speed of the ball at t = 3.4 s.
(b) What angle does make with the x axis at this moment?
(c) While the ball is in the air, at what value of t is the speed the smallest?
To find the answers to these questions, we need to use the equations of motion for projectile motion. In projectile motion, the horizontal and vertical motions are separate, so we will need to analyze them independently.
Given:
Initial speed (v₀) = 42 m/s
Angle (θ) = 30°
Time (t) = 3.4 s
Let's start by solving part (a):
(a) To find the speed of the ball at t = 3.4 s, we need to calculate the horizontal and vertical components of the velocity at that time.
First, let's find the horizontal component of the velocity (v₀x). This can be calculated using the equation:
v₀x = v₀ * cos(θ)
where v₀ is the initial speed and θ is the angle.
Plugging in the values, we get:
v₀x = 42 m/s * cos(30°)
v₀x = 42 m/s * √3/2
v₀x ≈ 36.372 m/s (rounded to three decimal places)
Next, let's find the vertical component of the velocity (v₀y). This can be calculated using the equation:
v₀y = v₀ * sin(θ)
Plugging in the values, we get:
v₀y = 42 m/s * sin(30°)
v₀y = 42 m/s * 1/2
v₀y = 21 m/s
Now, we need to find the speed of the ball at t = 3.4 s. Since the horizontal and vertical motions are independent, the speed at any given time can be found using the Pythagorean theorem:
speed = √(v₀x² + v₀y²)
Plugging in the values, we get:
speed = √((36.372 m/s)² + (21 m/s)²)
speed ≈ 42.24 m/s (rounded to two decimal places)
So, the speed of the ball at t = 3.4 s is approximately 42.24 m/s.
Now, let's move on to part (b):
(b) To find the angle the ball makes with the x-axis at t = 3.4 s, we can use the inverse tangent function (tan⁻¹).
The angle (θ') can be calculated using the equation:
θ' = tan⁻¹(v₀y / v₀x)
Plugging in the values, we get:
θ' = tan⁻¹(21 m/s / 36.372 m/s)
θ' ≈ 30.9°
So, the angle the ball makes with the x-axis at t = 3.4 s is approximately 30.9°.
Finally, let's solve part (c):
(c) To find the time at which the speed is the smallest, we need to consider that the vertical component of the velocity will be zero at the highest point of the ball's trajectory. At that point, the ball is momentarily at rest before it starts falling downwards.
We can find the time when the vertical component is zero (t₀) using the equation:
v₀y - gt₀ = 0
where g is the acceleration due to gravity. For simplicity, we will assume a value of g = 9.8 m/s².
Plugging in the values, we get:
21 m/s - 9.8 m/s² * t₀ = 0
t₀ = 21 m/s / (9.8 m/s²)
t₀ ≈ 2.143 s (rounded to three decimal places)
So, at approximately t = 2.143 s, the speed of the ball is the smallest.
Remember to always double-check your calculations and units to ensure accuracy.