Standing on a balcony, you throw your keys to a friend standing on the ground below. 1.2 seconds after you release the keys, they have an instantaneous velocity of 15.7 m/s, directed 52° below the horizontal. What initial velocity did you give them?
To determine the initial velocity with which you threw the keys, we can use the equations of motion for projectile motion. Let's break down the given information and solve step by step:
1.2 seconds after release:
- The keys have an instantaneous velocity of 15.7 m/s, directed 52° below the horizontal.
Step 1: Break the velocity into horizontal and vertical components.
The velocity has two components: horizontal (Vx) and vertical (Vy).
Vx represents the horizontal velocity.
Vy represents the vertical velocity.
Given that the keys were thrown at an angle below the horizontal, we can determine Vx and Vy using trigonometry:
Vx = V * cos(θ)
Vy = V * sin(θ)
Where:
V represents the initial velocity (what we need to find).
θ represents the angle of 52° below the horizontal.
Step 2: Calculate the horizontal and vertical components.
Using the given information, we can substitute the values into the equations:
Vx = V * cos(θ) = 15.7 m/s * cos(52°)
Vy = V * sin(θ) = 15.7 m/s * sin(52°)
Step 3: Solve for V.
We need to solve for V, the initial velocity. We can use the Pythagorean theorem:
V^2 = Vx^2 + Vy^2
Substituting the calculated values:
V^2 = (15.7 m/s * cos(52°))^2 + (15.7 m/s * sin(52°))^2
V^2 = (15.7 m/s)^2 * (cos(52°))^2 + (15.7 m/s)^2 * (sin(52°))^2
V^2 = (15.7 m/s)^2 * (cos^2(52°) + sin^2(52°))
V^2 = (15.7 m/s)^2 * 1
V^2 = (15.7 m/s)^2
V = √((15.7 m/s)^2)
V ≈ 15.7 m/s
Therefore, the initial velocity with which you threw the keys was approximately 15.7 m/s.
To find the initial velocity of the keys, we can use the equation for the horizontal component of velocity:
Vx = V * cos(theta)
where Vx is the horizontal component of velocity, V is the initial velocity, and theta is the angle below the horizontal.
Given that Vx = 15.7 m/s and theta = 52°, we can substitute these values into the equation to solve for V:
15.7 m/s = V * cos(52°)
Next, we need to find the vertical component of velocity, Vy. We can use the equation:
Vy = V * sin(theta)
where Vy is the vertical component of velocity.
To find Vy, we need to determine the time it takes for the keys to reach their instantaneous velocity. We know that time is 1.2 seconds.
Using the equation for vertical velocity with initial velocity Vy, acceleration due to gravity (g), and time (t), we have:
Vy = g * t
where g is the acceleration due to gravity, approximately 9.8 m/s². Substituting the given values into the equation, we have:
Vy = 9.8 m/s² * 1.2 s
Now we have both Vx and Vy. We can find the initial velocity V using the Pythagorean theorem:
V = sqrt(Vx² + Vy²)
Substituting the values we have calculated:
V = sqrt((15.7 m/s)² + (9.8 m/s² * 1.2 s)²)
Performing the calculations:
V = sqrt(246.49 m²/s² + 11.76 m²/s²)
V = sqrt(258.25 m²/s²)
V ≈ 16.08 m/s
Therefore, the initial velocity you gave the keys was approximately 16.08 m/s.