A cart rolls down a 2.5 m frictionless ramp in 1.25 s after starting from rest at the top. What is the angle of the ramp?
To find the angle of the ramp, we can use the principles of physics related to motion on inclined planes. Specifically, we will use the equation that relates the angle of the ramp to the acceleration of the cart.
The key information given in the question is that the cart rolls down the ramp in 1.25 s after starting from rest at the top. This implies that the initial velocity of the cart is zero.
The equation we will use is:
a = g * sin(θ)
Where:
a is the acceleration of the cart,
g is the acceleration due to gravity (approximately 9.8 m/s²),
θ is the angle of the ramp.
Since the cart is moving down the ramp, the acceleration is in the same direction as the ramp. Therefore, the acceleration will be positive.
First, let's calculate the acceleration using the given time:
a = Δv / t
Since the initial velocity (u) is zero, the equation simplifies to:
a = v / t
Where:
v is the final velocity of the cart,
t is the time taken.
The final velocity can be calculated using the formula:
v = u + a * t
Since the initial velocity is zero, the equation simplifies to:
v = a * t
Now, substituting the values given in the question:
v = a * t
v = a * 1.25
Thus, to find the acceleration, we divide both sides of the equation by 1.25:
a = v / 1.25
Now, using the previous equation from motion on inclined planes:
a = g * sin(θ)
We can substitute the value of a:
g * sin(θ) = v / 1.25
Rearranging the equation to solve for sin(θ):
sin(θ) = (v / 1.25) / g
Finally, to find the angle of the ramp, we take the inverse sine (sin⁻¹) of both sides:
θ = sin⁻¹((v / 1.25) / g)
Using the value of g (approximately 9.8 m/s²) and the final velocity obtained earlier, we can substitute these values and solve for θ, giving us the angle of the ramp.