Question

A 0.40-kg ball is thrown with a speed of 7.5 m/s at an upward

angle of 34 ∘. a) What is its speed at its highest point? b) How
high does it go? (Use conservation of energy.)

Answer 1

a) The speed of the ball at its highest point is 0 m/s. b) The ball goes approximately 5.74 meters high.

Step-by-step explanation:

a) To find the speed of the ball at its highest point, we can use the concept of conservation of energy. At the highest point, all of the ball's initial kinetic energy will be converted into potential energy. So, the speed at its highest point will be 0. We can calculate the initial kinetic energy using the formula: KE = 0.5 × mass × velocity². Plugging in the values, we get KE = 0.5 × 0.40 kg × (7.5 m/s)² = 22.5 J. So, at its highest point, the ball's speed is 0 m/s.

b) To find how high the ball goes, we can calculate the change in potential energy. The initial potential energy, when the ball is at the highest point, will be equal to the change in potential energy from the initial launch position. We can calculate the potential energy using the formula: PE = mass × gravity × height. Rearranging the formula, we get height = PE / (mass × gravity). Plugging in the values, we get height = 22.5 J / (0.40 kg × 9.8 m/s²) = 5.74 m. So, the ball goes approximately 5.74 meters high.

Answer 2

The speed at the highest point of the ball's trajectory is the horizontal component of its initial velocity. The maximum height is calculated using the conservation of energy, converting initial kinetic energy into gravitational potential energy.

Step-by-step explanation:

When a 0.40-kg ball is thrown with a speed of 7.5 m/s at an upward angle of 34 degrees, we are focused on examining both its vertical motion and the effects of conservation of energy to solve for its speed at the highest point and the maximum height it reaches.

At the highest point of its trajectory, the ball's vertical speed is zero since gravity has slowed down the upward motion to a stop before it begins to fall back down. However, the horizontal component of the velocity remains unchanged because there is no horizontal acceleration (ignoring air resistance). To find the horizontal component of the velocity, we use the cosine of the angle: speed_horizontal = 7.5 m/s * cos(34°). Only this horizontal component of speed will remain at the highest point.

To find out how high the ball goes, we utilize the principle of conservation of energy. The initial kinetic energy of the ball is converted into gravitational potential energy at the highest point: KE_initial = PE_top, with PE = m * g * h, where m is mass, g is acceleration due to gravity (9.8 m/s^2), and h is height. Using the initial vertical component of the speed (7.5 m/s * sin(34°)), we calculate the initial kinetic energy and solve for h.