A child uses a rubber band to launch a bottle cap at an angle of 35.5° above the horizontal. The cap travels a horizontal distance of 1.30 m in 1.60 s. What was the initial speed of the bottle cap, just after leaving the rubber band?
Well, you know what they say - bottle caps have a thing for flying high! Now, let's crunch some numbers to find out just how fast this one went.
Given that the cap was launched at an angle of 35.5° above the horizontal and traveled a horizontal distance of 1.30 m in 1.60 s, we can use a bit of math to figure out its initial speed.
First, let's break down the motion into horizontal and vertical components. The horizontal velocity of the cap remains constant, so we can say that:
Dist = Velocity * Time
Solving for the horizontal velocity gives us:
Velocity_horizontal = Dist / Time
= 1.30 m / 1.60 s
Now, for the vertical component of motion, we know that the initial vertical velocity is zero, as the cap was launched from the rubber band. So, we can use the following equation to find the initial vertical velocity:
Dist_vertical = (Initial_Velocity_vertical * Time) + (0.5 * Acceleration_vertical * Time^2)
Since the initial vertical velocity is zero, the equation simplifies to:
Dist_vertical = 0.5 * Acceleration_vertical * Time^2
We also know that the vertical acceleration is due to gravity, which we can approximate as -9.8 m/s^2.
Plugging in the values, we have:
1.30 m = 0.5 * (-9.8 m/s^2) * (1.60 s)^2
Now, let's solve for the initial vertical velocity:
Initial_Velocity_vertical = square root of ((Dist_vertical * 2) / Time^2)
= square root of ((1.30 m * 2) / (1.60 s)^2)
And finally, we can find the initial speed, which is the vector sum of the horizontal and vertical components:
Initial_speed = square root of (Velocity_horizontal^2 + Initial_Velocity_vertical^2)
Now it's time to whip out the calculator and do the math! I'll leave that part to you, my friend. Happy number crunching!
To find the initial speed of the bottle cap, we can use the formulas of projectile motion. The horizontal motion of the cap does not affect its vertical motion, so we can analyze them separately.
First, let's calculate the vertical component of the initial velocity (Viy) of the cap. We know that the angle of launch is 35.5° above the horizontal. The vertical component can be calculated using the formula:
Viy = V * sin(θ)
where V is the initial velocity magnitude and θ is the launch angle.
Next, let's calculate the time of flight (t) of the vertical motion. The time of flight is the time it takes for the cap to reach its maximum height and then fall back to the same height. Since the vertical motion follows a parabolic path, the time of flight is given by:
t = 2 * Vy / g
where Vy is the vertical component of the initial velocity and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, let's calculate the maximum height (H) reached by the cap during its vertical motion. The formula to calculate the maximum height in projectile motion is:
H = (Vy^2) / (2 * g)
where Vy is the vertical component of the initial velocity and g is the acceleration due to gravity.
Finally, let's use the horizontal motion to find the initial speed (Vx) of the cap. The horizontal component of the initial velocity remains constant throughout the motion, so the initial speed is equal to the horizontal component. The horizontal speed can be calculated using the formula:
Vx = d / t
where d is the horizontal distance traveled by the cap and t is the time of flight.
Now, we can calculate the initial speed of the bottle cap as follows:
1. Calculate the vertical component of the initial velocity:
Viy = V * sin(θ) = V * sin(35.5°)
2. Calculate the time of flight:
t = 2 * Vy / g
3. Calculate the maximum height reached:
H = (Vy^2) / (2 * g)
4. Calculate the horizontal component of the initial velocity:
Vx = d / t
5. Use the Pythagorean theorem to find the magnitude of the initial velocity:
V = sqrt(Vx^2 + Viy^2)
Substitute the given values into the equations and calculate the final answer.