In the latest Indian Jones film, Indy is supposed to throw a grenade from his car, which is going 93.0 km/h, to his enemy's car, which is going 130km/h. The enemy's car is 17.0m in front of the Indy's when he lets go of the grenade.
1)If Indy throws the grenade so its initial velocity relative to him is at an angle of 45above the horizontal, what should the magnitude of the initial velocity be? The cars are both traveling in the same direction on a level road. You can ignore air resistance.
2)Find the magnitude of the velocity relative to the earth.
To solve these problems, we can break them down into smaller parts and use the principles of relative motion. Let's go step by step:
First, we need to find the relative velocity of the enemy's car with respect to Indy's car. Given that the enemy's car is in front of Indy by 17.0m, and both cars are moving in the same direction, we can calculate the relative velocity as follows:
Relative Velocity = Velocity of Enemy's Car - Velocity of Indy's Car
Given that the velocity of the enemy's car is 130 km/h and the velocity of Indy's car is 93.0 km/h, we convert these speeds to meters per second:
Velocity of Enemy's Car = 130 km/h = (130 * 1000) / 3600 m/s ≈ 36.11 m/s
Velocity of Indy's Car = 93.0 km/h = (93.0 * 1000) / 3600 m/s ≈ 25.83 m/s
Relative Velocity = 36.11 m/s - 25.83 m/s = 10.28 m/s
Now, let's move on to the first question:
1) If Indy throws the grenade, its initial velocity relative to him is at an angle of 45 degrees above the horizontal. We need to find the magnitude of the initial velocity.
Let's consider the horizontal and vertical components of the initial velocity:
Horizontal Component: Since the angle is 45 degrees, the horizontal component of the initial velocity will be the same as the vertical component.
Vertical Component: We know that the relative velocity of the enemy's car with respect to Indy's car is 10.28 m/s. So, the vertical component of the initial velocity should be equal to the relative velocity.
Therefore, the magnitude of the initial velocity can be calculated using the Pythagorean theorem:
Magnitude of Initial Velocity = sqrt((Horizontal Component)^2 + (Vertical Component)^2)
Since the angle is 45 degrees, let's assume the magnitude of the initial velocity as 'v.'
Horizontal Component = v * cos(45 degrees) ≈ 0.707 * v
Vertical Component = v * sin(45 degrees) ≈ 0.707 * v
Now, using the Pythagorean theorem, we have:
v = sqrt((0.707 * v)^2 + (10.28 m/s)^2)
Simplifying the equation:
v = sqrt(0.5 * v^2 + 105.6284 m^2/s^2)
Squaring both sides of the equation:
v^2 = 0.5 * v^2 + 105.6284 m^2/s^2
Rearranging the equation:
0.5 * v^2 = 105.6284 m^2/s^2
v^2 = 211.2568 m^2/s^2
v ≈ sqrt(211.2568) m/s ≈ 14.54 m/s
Therefore, the magnitude of the initial velocity should be approximately 14.54 m/s.
2) To find the magnitude of the velocity relative to the Earth, we need to consider the velocity of Indy's car.
Since both cars are moving in the same direction, the velocity relative to the Earth is the sum of the velocity of Indy's car and the relative velocity:
Velocity Relative to Earth = Velocity of Indy's Car + Relative Velocity
Velocity Relative to Earth ≈ 25.83 m/s + 10.28 m/s ≈ 36.11 m/s
Therefore, the magnitude of the velocity relative to the Earth is approximately 36.11 m/s.