A daredevil jumps a canyon 12 m wide . To do so , he drives a car up a 19 degree incline.

a) What is the minimum speed must he achieve to clear the canyon ?

b) If the daredevil jumps at this minimum speed, what will his speed be when he reaches the other side ?

a) Well, jumping a canyon is no small feat, especially if you're driving up an incline! To clear the 12 m wide canyon, the daredevil must achieve a minimum speed. Let's calculate that:

We can use the formula for projectile motion:

Range = (v^2 * sin(2θ)) / g

Where:
- Range is the distance traveled horizontally,
- v is the initial velocity,
- θ is the angle of inclination (19 degrees),
- and g is the acceleration due to gravity.

Given that the canyon is 12 m wide and the angle of inclination is 19 degrees, we can rearrange the formula to find the minimum speed:

12 = (v^2 * sin(2 * 19)) / g

Solving for v, we have:

v = √((12 * g) / sin(38))

Using the acceleration due to gravity, g, which is approximately 9.8 m/s^2, we can calculate the minimum speed the daredevil must achieve.

b) Jumping across a canyon requires quite a bit of speed, but what about when the daredevil reaches the other side? Let's find out!

Since the daredevil jumps with the minimum speed required to clear the 12 m wide canyon, we can assume that he maintains a constant speed throughout the jump. Therefore, his speed when he reaches the other side will be the same as his minimum speed.

So, the daredevil's speed when he reaches the other side will be the same as the minimum speed we calculated in part (a).

Now, remember, this is all assuming the daredevil doesn't make any pit stops for a snack or a quick nap along the way!

To solve this problem, we can use the principles of projectile motion. Let's break down the steps for both parts of the question:

a) To clear the canyon, the daredevil's car must follow a projectile motion trajectory. We need to find the minimum speed he must achieve to clear the canyon.

Step 1: Find the vertical displacement (height) of the canyon.
In this case, the vertical displacement is zero since the daredevil's car starts and ends at the same height.

Step 2: Find the horizontal distance (range) of the canyon.
Given: Width of the canyon = 12 meters
The range of projectile motion is given by the equation:
Range = (initial velocity)^2 * sin(2θ) / g
Where θ is the angle of the incline and g is the acceleration due to gravity (approximately 9.8 m/s^2)

Rearranging the equation, we get:
(initial velocity)^2 = Range * g / sin(2θ)

Plugging in the values:
Range = 12 m
θ = 19 degrees
g = 9.8 m/s^2

(initial velocity)^2 = (12 m) * (9.8 m/s^2) / sin(2 * 19 degrees)

Solving for (initial velocity)^2:
(initial velocity)^2 = 67.4921
(initial velocity) ≈ √67.4921
(initial velocity) ≈ 8.21 m/s

b) To find the speed when the daredevil reaches the other side, we can use the conservation of energy principle. The total mechanical energy (kinetic energy + potential energy) is conserved.

Step 1: Find the potential energy at the starting point.
The potential energy (PE) at the starting point is given by the equation:
PE = mass * gravity * height
Since the height is the same at both the starting and ending points, the potential energy is equal to zero.

Step 2: Find the kinetic energy at the ending point.
The kinetic energy (KE) at the ending point is given by the equation:
KE = 0.5 * mass * velocity^2

Setting the potential energy equal to the kinetic energy:
PE = KE
0 = 0.5 * mass * velocity^2

Solving for velocity:
velocity^2 = 0 / (0.5 * mass)
velocity = 0

Therefore, the daredevil's speed when he reaches the other side will be zero since all the potential energy is converted to kinetic energy during the jump.

To solve this problem, we can break it down into two main parts:

1) Finding the minimum speed required to clear the canyon.
2) Finding the speed of the daredevil when he reaches the other side of the canyon.

Let's start with the first part:

1) Finding the minimum speed required to clear the canyon:

To clear the canyon, the daredevil's car needs to have enough horizontal velocity when it leaves the incline so that it can travel the full width of the canyon. In other words, the horizontal component of the car's velocity must be equal to or greater than the width of the canyon.

Since the width of the canyon is 12m, we need to find the horizontal component of the car's velocity.

To find the horizontal component, we can use trigonometry. The horizontal component is given by the formula:

horizontal component = velocity * cos(angle)

Here, the angle is given as 19 degrees and we need to find the velocity.

Using trigonometry, cos(19 degrees) ≈ 0.943

So, the horizontal component of velocity = velocity * 0.943

To clear the canyon, the horizontal component of the car's velocity must be equal to or greater than 12m. So we can set up the equation:

velocity * 0.943 ≥ 12

Solving for velocity:

velocity ≥ 12 / 0.943

velocity ≥ 12.73 m/s

Therefore, the minimum speed the daredevil must achieve to clear the canyon is approximately 12.73 m/s.

Now, let's move on to the second part:

2) Finding the speed of the daredevil when he reaches the other side of the canyon:

To find the speed of the daredevil when he reaches the other side of the canyon, we need to consider the conservation of energy.

Assuming no energy losses due to friction or air resistance, the initial potential energy of the daredevil's car is converted into kinetic energy on reaching the other side of the canyon.

Mathematically, we can write this as:

Initial potential energy = Final kinetic energy

The initial potential energy is given by:

Initial potential energy = m * g * h

Here, m represents the mass of the car, g represents the acceleration due to gravity (approximately 9.8 m/s^2), and h represents the height of the incline.

Since we don't have the mass of the car, we can cancel it out by dividing both sides of the equation by the mass. So, we are left with:

g * h = 1/2 * velocity^2

Rearranging the equation to solve for velocity:

velocity = √(2 * g * h)

Substituting the values, g = 9.8 m/s^2 and h = 12 m * sin(19 degrees):

velocity = √(2 * 9.8 * 12 * sin(19 degrees))

velocity ≈ 12.95 m/s

Therefore, the speed of the daredevil when he reaches the other side of the canyon, given that he jumps at the minimum speed to clear the canyon, is approximately 12.95 m/s.