1.A firetruck is traveling at a velocity of +20 m/s in the x direction. As the truck passes x=63m it shoots a tennis ball backwards with a speed of 20 m/s relative to the truck at an angle of 39° from a height of 1.4 m. Neglect air resistance. At what x value does the tennis ball hit the ground?

2.The firetruck goes around a 180°, 162 m radius circular curve. It enters the curve with a speed of 12.6 m/s and leaves the curve with a speed of 38.8 m/s. Assuming the speed changes at a constant rate, what is the magnitude of the total acceleration of the firetruck just after it has entered the curve?

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To solve these questions, we will need to use some principles of physics and mathematical equations. Let's break down each question and go step by step to find the answers.

1. To determine the x value where the tennis ball hits the ground, we need to analyze the motion of the ball separately from the firetruck's motion. Here's how we can approach it:

Step 1: Determine the time taken by the tennis ball to hit the ground.
- Since we are neglecting air resistance, we can consider the vertical motion of the ball as a projectile motion.
- The initial vertical velocity of the ball is given by V_y = V * sin(θ), where V is the speed of the ball relative to the truck and θ is the launch angle.
- The vertical motion is influenced by gravity, so the equation for the displacement in the y direction can be written as y = V_y*t - (1/2)*g*t^2, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- At the moment the ball hits the ground, y = 0. We can solve this equation to find the time it takes for the ball to hit the ground.

Step 2: Determine the corresponding x value when the ball hits the ground.
- The horizontal motion of the ball is independent of its vertical motion and unaffected by gravity.
- The horizontal velocity of the ball is given by V_x = V * cos(θ), where V is the speed of the ball relative to the truck and θ is the launch angle.
- The distance covered by the ball in the x direction can be calculated using x = V_x * t, where t is the time determined in Step 1.

By applying these steps, you can find the x value where the ball hits the ground.

2. To determine the magnitude of the total acceleration of the firetruck just after it has entered the curve, we can use the principles of circular motion:

Step 1: Determine the acceleration due to the change in speed.
- The given information states that the firetruck enters the curve with a speed of 12.6 m/s and leaves with a speed of 38.8 m/s.
- The change in velocity can be calculated by subtracting the initial velocity from the final velocity, Δv = v_f - v_i.
- The time taken for the change in speed is not given, so we cannot directly calculate acceleration (acceleration, a = Δv / t). However, we can use another equation for acceleration.

Step 2: Apply the equation for acceleration in circular motion.
- The formula for acceleration in circular motion is a = v^2 / r, where v is the velocity and r is the radius of the curve.
- The acceleration for a given velocity is always directed towards the center of the curve.
- Since the firetruck is moving on a circular curve, the magnitude of the total acceleration is given by the sum of the centripetal acceleration and the tangential acceleration.

By applying these steps, you can calculate the magnitude of the total acceleration of the firetruck.