For a scene in a movie, a stunt driver drives a 1.50×10^3kg SUV with a length of 4.00m around a circular curve with a radius of curvature of 0.333 km. The vehicle is to be driven off the edge of a gully 10.0 m wide, and land on the other side 2.96 m below the initial side.
What is the minimum centripetal acceleration the SUV must have in going around the circular curve to clear the gully and land on the other side?
This is how I attempted the problem: please tell me what I did wrong.
y = 1/2*g*t^2 or t= sqrt(2*y/g) = sqrt(2*2.96/9.80) = 0.777s
So the speed needed to cross the gully is vx = x/t = 10.0m/0.777s = 12.87m/s
Now the centripetal acceleration would be v^2/r = 12.87^2/333m= 0.497m/s^2
Nevermind, I realized my mistake. I forgot to add the length of the SUV (4.00 m) to 10.0 m.
Your approach to the problem is partially correct, but you made a mistake in calculating the centripetal acceleration.
Let's start with the correct calculation of the time it takes for the vehicle to cross the gully:
Using the equation y = 1/2 * g * t^2 for vertical motion, we have:
y = 2.96 m (vertical distance)
g = 9.8 m/s^2 (acceleration due to gravity)
Rearranging the equation, we get:
t^2 = 2y/g
t^2 = 2 * 2.96 / 9.8
t^2 = 0.6041
t = √0.6041s
t ≈ 0.7788s
The speed needed to cross the gully is calculated correctly as:
vx = x/t
vx = 10.0 m / 0.7788 s
vx ≈ 12.85 m/s
Now, let's calculate the minimum centripetal acceleration required to clear the gully and land on the other side.
Centripetal acceleration is given by the formula:
ac = v^2 / r
where:
v = velocity
r = radius of curvature
In this case, the velocity is given as vx = 12.85 m/s, but we need to convert the radius of curvature from kilometers to meters.
radius = 0.333 km * 1000 m/km
radius = 333 m
Now, we can calculate the centripetal acceleration:
ac = vx^2 / r
ac = (12.85 m/s)^2 / 333 m
ac = 0.494 m/s^2
So, the correct minimum centripetal acceleration the SUV must have in going around the circular curve to clear the gully and land on the other side is approximately 0.494 m/s^2.
In your attempt to solve the problem, you correctly used the formula for vertical displacement to find the time it takes for the SUV to cross the gully. You correctly found t = 0.777s.
However, the next step is where you made a mistake. The formula you used to find the speed needed to cross the gully, vx = x/t, assumes constant velocity. In reality, the SUV will be accelerating because it needs to maintain a curved path around the circular curve. So we cannot use this formula to find the required speed.
To find the minimum centripetal acceleration the SUV must have to clear the gully and land on the other side, we need to consider the forces acting on the SUV.
The SUV will experience two forces: the gravitational force (mg) pulling it downwards, and the normal force (N) exerted by the ground pushing it upwards. These two forces must cancel each other out to maintain equilibrium.
Since the SUV is in circular motion, the net force acting on it must be directed towards the center of the curve (centripetal force).
The centripetal force is given by the equation Fc = m * ac, where m is the mass of the SUV and ac is the centripetal acceleration.
To find the minimum centripetal acceleration needed, we need to find the minimum centripetal force needed. This is because the minimum velocity needed to clear the gully and land on the other side will correspond to the minimum force needed.
The minimum force needed can be found using the normal force, N. N can be calculated using the equation N = mg - Fg, where Fg is the gravitational force pulling the SUV downwards.
Fg = mg, since gravitational force is proportional to mass.
Then, N = mg - mg = 0. The normal force is zero because there is no ground pushing the SUV upwards when it is in mid-air.
Since N = 0, the minimum centripetal force required to maintain equilibrium is also zero.
Therefore, the minimum centripetal acceleration required is also zero.
Thus, the SUV does not need any minimum centripetal acceleration to clear the gully and land on the other side.