Question

A ball is thrown at an angle of 45° to the ground. If the ball lands 76 m away, what was the initial speed of the ball?

Answer 1

Using the range equation for projectile motion, and given the range of 76 m and angle of 45°, the initial speed of the ball is calculated to be approximately 27.4 m/s when gravity is 9.81 m/s² and air resistance is ignored.

Step-by-step explanation:

To calculate the initial speed of a ball thrown at a 45° angle and landing 76 m away, we use the projectile motion equations. Assuming the acceleration due to gravity is 9.81 m/s2, no air resistance, and the ball lands at the same height from which it was thrown:

The range equation for projectile motion is R = (v2 sin(2θ)) / g, where R is the range (76 m), v is the initial velocity, θ is 45°, and g is 9.81 m/s2. Since sin(90°) = 1, the equation simplifies to R = v2 / g.

• Solving for v gives us v = sqrt(R × g).
• Substitute R = 76 m and g = 9.81 m/s2 into the equation.
• v = sqrt(76 m × 9.81 m/s2)
• v ≈ 27.4 m/s.

The initial speed of the ball is approximately 27.4 m/s.

Answer 2

The initial speed of a ball thrown at a 45° angle that lands 76 m away can be found using the projectile motion range equation, resulting in the calculation of the square root of 74.8 times 9.8 m/s.

Step-by-step explanation:

To determine the initial speed of a ball thrown at an angle of 45° to the ground that lands 76 m away, we can use the projectile motion equations. Since the angle is 45°, the horizontal and vertical components of the initial velocity are equal. Furthermore, because the range (R) is given, we can utilize the range equation for projectile motion on level ground:

R = (v^2 \times sin(2\theta)) / g, where v is the initial speed, \theta is the angle of projection, and g is the acceleration due to gravity (9.8 m/s^2).

In this case, \theta equals 45°, and sin(2\times45°) is 1. Plugging in the values we get:

76 = (v^2 \times 1) / 9.8

v^2 = 76 \times 9.8

v = sqrt(76 \times 9.8)

The initial speed v can be calculated by taking the square root of 74.8 times 9.8, which will give us the result in meters per second (m/s).