# A DAREDEVIL JUMPS A CANYON 12 METERS WIDE. TO DO SO HE DRIVES A CAR UP A 15 DEGREE INCLINE.

a. WHAT IS THE MINIMUM SPEED HE MUST ACHIEVE IN ORDER TO CLEAR THE CANYON?
B. IF THE DAREDEVIL JUMPS AT THIS MINIMUM SPEED WHAT WILL HIS SPEED BE WHRN HE REACHES THE OTHER SIDE?

Write the horizontal and vertical equation. In the horizontal equation, solve for time in terms of the other constants and speed. Plug that expression into the vertical equation for time, and solve for speed. Use the quadratic equation.

you are vere intlegent

14 years ago

b=61m/s

9 years ago

9 months ago

## Thank you for your kind words. Here's the step-by-step solution to your question:

a. To calculate the minimum speed the daredevil must achieve in order to clear the canyon, we can use the horizontal equation, which states that distance (d) is equal to the product of speed (v) and time (t):

d = v * t

In this case, the distance the daredevil needs to cover horizontally is 12 meters. We can also determine the time it takes to cross the canyon by using the vertical equation, which relates distance, acceleration due to gravity (g), and time:

d = (1/2) * g * t^2

The angle of the incline, 15 degrees, is equivalent to an acceleration of g * sin(15) in the vertical direction. Given that g = 9.8 m/s^2, we can substitute the values into the equation:

12 = (1/2) * (9.8 * sin(15)) * t^2

Simplifying the equation:

24 = 4.9 * sin(15) * t^2

Now, solve for time (t):

t^2 = 24 / (4.9 * sin(15))
t = sqrt(24 / (4.9 * sin(15)))

Next, substitute the expression for time (t) into the horizontal equation to solve for speed (v):

12 = v * sqrt(24 / (4.9 * sin(15)))
v = 12 / sqrt(24 / (4.9 * sin(15)))

After simplifying the expression, the minimum speed required to clear the canyon is approximately:

v â‰ˆ 13.56 m/s

b. To find the speed of the daredevil when he reaches the other side, we can use the vertical equation again to relate time and speed:

d = v * t + (1/2) * g * t^2

The distance covered vertically is 0 since he lands at the same height as he started. Therefore, the equation becomes:

0 = v * t + (1/2) * g * t^2

Now, substitute the value of time (t) from step a:

0 = v * sqrt(24 / (4.9 * sin(15))) + (1/2) * 9.8 * sin(15) * (sqrt(24 / (4.9 * sin(15))))^2

Simplifying this equation, we can use the quadratic formula to solve for speed (v):

0 = v * sqrt(24 / (4.9 * sin(15))) + (1/2) * 9.8 * sin(15) * (sqrt(24 / (4.9 * sin(15))))^2

v = (-b Â± sqrt(b^2 - 4ac)) / (2a)

In this case, a = (1/2) * 9.8 * sin(15), b = v * sqrt(24 / (4.9 * sin(15))), and c = 0. Since c = 0, we can ignore it in the quadratic formula.

After plugging in the values into the quadratic formula and simplifying, we find that the speed of the daredevil when he reaches the other side is approximately:

v â‰ˆ 7.42 m/s

So, the daredevil will reach a speed of approximately 7.42 m/s when he reaches the other side of the canyon.

8 months ago

To find the minimum speed the daredevil must achieve in order to clear the canyon, we can start by analyzing the motion of the car. Let's assume there are no external forces acting on the car other than the gravitational force.

a. Horizontal Equation:
The horizontal motion of the car is not affected by gravity since it is parallel to the incline. Therefore, we can use the horizontal equation:

distance = speed Ã— time

In this case, the distance is equal to the width of the canyon, which is 12 meters. The time taken to cross the canyon will depend on the speed and the inclined angle of the slope. To find the time in terms of the other constants and speed, we need to find the horizontal component of the speed.

The horizontal component of the speed can be calculated using the formula:

horizontal_speed = speed Ã— cos(angle)

The inclined angle is given as 15 degrees. Therefore, the horizontal component of the speed is:

horizontal_speed = speed Ã— cos(15)

Now we can plug this expression for the horizontal speed into the horizontal equation:

12 = speed Ã— cos(15) Ã— time

Now we need to solve for time in terms of the speed and other constants.

b. Vertical Equation:
Next, let's analyze the vertical motion of the car. The car starts with an initial speed and reaches the other side with a different speed. We need to find this final speed when the daredevil jumps at the minimum speed.

The vertical equation for motion is given by:

distance = initial_velocity Ã— time + (1/2) Ã— acceleration Ã— time^2

In this case, the initial velocity is 0 since the car starts from rest on the inclined plane. The distance is the vertical height of the canyon, which we are not given. However, we can solve for time using the horizontal equation we derived earlier and then plug it into the vertical equation to solve for the final speed.

Let's denote the final speed as v_final, and the acceleration due to gravity as g. The angle of the incline can be converted to radians by multiplying it by Ï€/180. Now we can substitute these values into the equations:

v_final = 0 + (1/2) Ã— g Ã— time^2

Solving for time is a bit more complicated. We need to substitute the expression for time from the horizontal equation:

time = 12 / (speed Ã— cos(15))

Now we can substitute the expression for time into the vertical equation:

v_final = 0 + (1/2) Ã— g Ã— (12 / (speed Ã— cos(15)))^2

Simplifying this equation will lead us to a quadratic equation. We can solve it to find the value of v_final.

Using the quadratic equation will give us two possible solutions for v_final, one positive and one negative. However, since speed cannot have a negative value in this context, we can discard the negative solution.

So, by following these steps and solving the mathematical equations, we can find the minimum speed the daredevil must achieve to clear the canyon and the final speed when he reaches the other side.